Integrable triples in semisimple Lie algebras

We classify all integrable triples in simple Lie algebras, up to equivalence. The importance of this problem stems from the fact that for each such equivalence class one can construct the corresponding integrable hierarchy of bi-Hamiltonian PDE. The simplest integrable triple $(f,0,e)$ in $\mathfrak{sl}_2$ corresponds to the KdV hierarchy, and the triple $(f,0,e_\theta)$, where $f$ is the sum of negative simple root vectors and $e_\theta$ is the highest root vector of a simple Lie algebra, corresponds to the Drinfeld-Sokolov hierarchy.


Introduction
The present paper is a continuation of our paper [DSJKV20], where we studied integrability of Walgebras. Namely, we showed that, for the classical affine W -algebra W(g, f ) attached to a simple Lie algebra g and its non-zero nilpotent element f , the Lie algebra W(g, f )/∂W(g, f ) contains an infinitedimensional abelian subalgebra, except, possibly, for the following f (in the notation of [CMG93]): 4A 1 , 2A 2 + 2A 1 , 2A 3 , A 4 + A 3 and A 7 in E 8 ; A 2 + A 1 in F 4 ; A 1 in G 2 . Consequently, for all these W -algebras (with the seven exceptions above) one constructs an integrable hierarchy of bi-Hamiltonian PDE, the simplest being the KdV hierarchy, constructed for W(sl 2 , f ).
The proof of this theorem consists of the following ingredients. First, it is the Drinfeld-Sokolov [DS85] (abbreviated DS) method of constructing integrals of motion in involution in the case when f is a principal nilpotent element of g, which has been extended to the case when f is a nilpotent element of semisimple type for g in [DSKV13].
Second, we showed in [DSJKV20] that a simple modification of the DS method works also for f of non-nilpotent type, which covers all nilpotent elements of even depth.
Third, in the same paper we showed that for f of nilpotent type this modification works as well, except for the seven nilpotent conjugacy classes mentioned above.
Let us now introduce the relevant definitions. By the Jacobson-Morozov theorem, any non-zero nilpotent f of a simple Lie algebra g can be included in an sl 2 -triple s = {e, f, h}. This produces a Z-grading of g by eigenspaces of ad h: (1.1) which, up to conjugation, is independent of the choice of the sl 2 -triple. The maximal j > 0 for which g j = 0 is denoted by d, and is called the depth of f . An element of the form f + E, where E is a non-zero element of g d , is called a cyclic element associated to f . The key tool in the DS method is Kostant's theorem that for the principal nilpotent f all the associated cyclic elements are semisimple. The main result of [DSKV13] extends the DS method in the framework of Poisson vertex algebras, to arbitrary f , which admits an associated semisimple cyclic element. Such f is called a nilpotent element of semisimple type [EKV13].
If all cyclic elements, associated to f , are nilpotent, then f is called a nilpotent element of nilpotent type. It is proved in [EKV13,Theorem 1.1] that f is of nilpotent type if and only if its depth is an odd integer. As has been mentioned above, the DS method works for f of non-nilpotent type.
In the case of f of nilpotent type, i. e. of odd depth d, we introduced in [DSJKV20] the notion of a quasi-cyclic element f + E, associated to f . It requires E to be a non-zero element of g d−1 with the additional requirement that the centralizer of E in g 1 is coisotropic with respect to the symplectic form ω on g 1 , defined by (1.2) Hereafter (· | ·) is a fixed non-degenerate symmetric invariant bilinear form on g. We show in [DSJKV20] that the DS method works for f of nilpotent type, provided that there exists a non-nilpotent quasi-cyclic element, associated to f . Moreover, such an element exists if and only if f is not one of the seven nilpotent elements of odd depth mentioned above. In the present paper we study quasi-cyclic elements, associated to arbitrary non-zero nilpotent elements in simple Lie algebras.
The main tool of the proof of the Integrability Theorem of [DSJKV20] is the notion of an integrable triple associated to the nilpotent element f . It is a triple of elements (f 1 , f 2 , E), where f 1 , f 2 ∈ g −2 , E ∈ g j with j 1, such that (I1) f = f 1 + f 2 and [f 1 , f 2 ] = 0; (I2) [E, g 2 ] = 0 and the centralizer l ⊥ of E in g 1 is coisotropic with respect to the symplectic form ω, defined by (1.2); (I3) f 1 + E is semisimple and [f 2 , E] = 0.
The coisotropy condition (I2) is important for the construction of the corresponding to f classical affine W -algebra, and condition (I3) is used for the construction of integrals of motion in involution.
Note that for an integrable triple (f 1 , f 2 , E) the decomposition f + E = (f 1 + E) + f 2 is the Jordan decomposition, and that E commutes with the subalgebra n := l ⊥ + g 2 .
It turns out that for j < d − 1 there are no integrable triples (Proposition 3.5). Obviously, for an integrable triple (f 1 , f 2 , E), associated to f , the element f + E = f 1 + f 2 + E is a cyclic (resp. quasicyclic) element, associated to f , if j = d (resp. j = d − 1). Such a cyclic (resp. quasi-cyclic) element is called integrable. In this case the element E ∈ g j is called integrable for f . Conversely, given a cyclic or quasi-cyclic element f + E, we can obtain an integrable triple associated to f , by taking the Jordan decomposition of f + E, provided that E is integrable for f . One of the main problems is when such an integrable triple exists.
In [DSJKV20] we have established existence of integrable triples, associated to any nilpotent element f , except for the seven cases mentioned above. In the present paper we give a complete solution to this problem. Namely, we find all integrable cyclic and quasi-cyclic elements f + E, up to equivalence, for each non-zero nilpotent element f in a simple Lie algebra g. Two integrable cyclic or quasi-cyclic elements f + E and f + E are called equivalent if E is proportional to an element from the orbit Z(s)(E), where Z(s) stands for the centralizer of the sl 2 -triple s in the adjoint group G of g. The importance of this problem stems from the fact that, as established in [DSJKV20], for each equivalence class of integrable triples one can construct the associated integrable hierarchy of bi-Hamiltonian PDE.
A partial solution to this problem was given in [EJK20], where all semisimple cyclic elements have been classified (the corresponding integrable triple has the form (f, 0, E)). The key observation there (checked by case-wise verification) was that the linear reductive group Z(s)|g d is polar. Polar linear groups were introduced in [DK85] as reductive linear groups, having properties, similar to the adjoint group (see Section 2.1 for the precise definition). This observation allows one to reduce considerations to the case E ∈ C, a Cartan subspace of g d , since it was proved in [EKV13], Proposition 2.2(a), that for semisimple cyclic element f + E, the orbit Z(s)(E) must be closed.
In the present paper we find that, remarkably, the linear reductive group Z(s)|g d−1 is polar as well! This is Theorem 2.8. Unfortunately we still need a case-wise analysis in its proof. However Remark 2.9 explains why Z(s)|g j should be polar for j = d and d − 1. Note that Z(s)|g j is not polar in general for 1 j < d − 1.
Our first main result is Theorem 3.6, which states that a cyclic element f + E is not nilpotent if and only if the Zariski closure of the orbit Z(s)(E) does not contain 0. Our second main result is Theorem 3.15, which states that a cyclic element f + E is integrable if and only if the orbit Z(s)(E) is closed. Such E are classified by a Cartan subspace of Z(s)|g d . A similar result holds for a quasi-cyclic element f + E, where E ∈ g d−1 and d is odd; in the case of even d we were able to prove only the "only if" part of this result (see Theorem 3.16).
These results allow us to classify completely all integrable cyclic and quasi-cyclic elements in all simple Lie algebras. As has been mentioned above, this is equivalent to the classification of integrable triples. For exceptional g this is done using the GAP package SLA by W. de Graaf [SLA], and the results are listed in Table 5 and Tables 3, 4, which represent the cases of d odd and even respectively. For classical g this is done by explicit calculations in the standard representation of sl N , sp N and so N . In particular, we have the following results on existence of integrable cyclic and quasi-cyclic elements in simple Lie algebras g, associated to non-zero nilpotent elements f ∈ g. (i) f has odd depth and is of the following type in exceptional g: 3A 1 in E 6 and E 8 ; 3A 1 in E 7 ; 2A 2 + A 1 in E 6 , E 7 , E 8 ; 4A 1 in E 7 ; A 1 +Ã 1 in F 4 , (ii) f has even depth and is of the following type in exceptional g: A 1 in all exceptional g; 2A 1 in E 6 ; A 2 + A 1 in E 6 , E 7 , E 8 ; A 4 + A 1 in E 6 , E 7 , E 8 ; iii) all f of odd depth in classical g, which happens only for g = so N , and the partition, corresponding to f , has odd largest part p 1 of multiplicity 1 and the next part p 2 = p 1 − 1, (iv) the following f of even depth, corresponding to the partition (p (r1) 1 , p (r2) 2 , ...) in classical g: g = sl N : r 1 r 2 ; g = so N : p 1 is even, r 1 = 2, p 2 = p 1 − 1, r 2 2.
Many statements in the paper are established by a case-wise verification. It would be very interesting to find more conceptual proofs. Here are some of these statements.
(1) The linear groups Z(s)|g j for j = d and d − 1 are polar.
(2) There exists at most one, up to equivalence, quasi-cyclic element for each nilpotent element f of even depth. (3) There are no integrable triples (f 1 , f 2 , E) with E ∈ g j , where j < d − 1. Throughout the paper the base field F is an algebraically closed field of characteristic 0. Though the theory of polar linear groups was developed in [DK85] over C, all results hold over F by the Lefschetz principle.
Acknowledgments. The first author was partially supported by the national PRIN 2017 grant "Moduli and Lie Theory" number 2017YRA3LK_001, and by the University 2019 grant "Vertex algebras and integrable Hamiltonian systems". The second author was partially supported by the grant FR-18-10849 of Shota Rustaveli National Science Foundation of Georgia. The third author was partially supported by the Bert and Ann Kostant fund, and by the Simons collaboration grant.
We would like to thank J. Dadok, A. Elashvili, P. Etingof, and D. Panyushev for very useful discussions. Extensive use of the computer algebra system GAP, and the package SLA by Willem de Graaf in particular, is gratefully acknowledged.
2. Polar linear groups and gradings for nilpotent elements 2.1. Review of polar linear groups. Let G be a reductive algebraic group, acting linearly and faithfully on a finite-dimensional vector space V , which will be denoted by G|V . It is well known that the subalgebra F[V ] G of G-invariant polynomials is finitely generated, hence the inclusion F[V ] G → F[V ] induces the map of the corresponding affine varieties It is well known that the map π is surjective and that each of the fibers of π contains a unique closed G-orbit (the orbit of minimal dimension in the fiber). The fiber over π(0), called the zero fiber, consists of elements v ∈ V , such that the Zariski closure of the orbit G(v) of v contains 0. Such elements are called nilpotent elements of the linear group G|V . Elements v ∈ V , such that G(v) is closed, are called semisimple elements of G|V . This terminology is motivated by the well-known fact that for the adjoint linear group G|g an element is semisimple (resp. nilpotent) if and only if it is a semisimple (resp. nilpotent) element of the Lie algebra g.
An efficient way of constructing semisimple elements for a reductive linear group G is given by the following Proposition 2.1 ( [DK85]). Let P be a set of weights of the g-module V from its irreducible components with the following properties: (i) λ i − λ j is not a root of g if i = j and λ i , λ j ∈ P are weights of the same irreducible component of V ; (ii) zero is an interior point of the convex hull of P . Let v λi , λ i ∈ P , be linearly independent weight vectors from irreducible components of V . Then the vector i v λi is semisimple. Proof. The proposition is slightly stronger than Proposition 1.2 from [DK85], but its proof is the same.
Corollary 2.2 (Kostant theorem). Any vector from the zero weight space of V is semisimple. Now we turn to the discussion of the especially nice class of reductive linear groups G|V , called polar linear groups, which were introduced in [DK85]. Let v ∈ V be a semisimple element, and let (2.2) and in this case C v is called a Cartan subspace of V . Note that the Cartan subalgebra h of the Lie algebra g of G is a Cartan subspace for the adjoint linear group G|g, since for a regular v ∈ h its G-orbit is closed and (2.4) holds because dim More generally, we have the following Proposition 2.3 ([EJK20]). Let G|V be a reductive linear group and let C be its zero weight space. Then and in the case of equality the linear group G|V is polar.
Remark 2.4. By the definition, G|V is polar if dim V / /G = 1, or dim V / /G = 0. Note also that the direct The following theorem shows that a Cartan subspace of a polar linear group G|V has the same basic properties as a Cartan subalgebra of g.
Then G(C) consists of all semisimple elements of V , and for any semisimple v ∈ V , the orbit G(v) intersects C by a (non-empty) orbit of W .
Recall that a linear reductive group G|V is called stable if V has a non-empty Zariski open subset, consisting of closed G-orbits. The following proposition is very useful in verifying that a stable linear reductive group G|V is polar. Proposition 2.6 ( [DKII]). Let G|V be a stable reductive linear group. Let C ⊂ V be a subspace, such that V = g(C) ⊕ C (2.6) and (2.7) Then G|V is polar and C is a Cartan subspace.
Proof. Due to (2.6), G(C) contains a Zariski open subset, and, due to stability, it contains a Zariski open subset Ω consisting of closed G-orbits of maximal dimension. Let C • = C ∩ Ω, and let v ∈ C • . Then the tangent space T v to G(v) at v lies in g(C), and, due to (2.6) and (2.7), actually T v = g(C). Since this holds for all v ∈ C • , we obtain that C v = C. Hence, by (2.7), G|V is polar and C is a Cartan subspace.
Remark 2.7. (a) It follows from Theorem 2.5 (a) that conditions (2.6) and (2.7) are also necessary for a reductive linear group to be stable polar.
(b) By Popov's stability theorem [Po70], for a semisimple algebraic group G stability of a linear group G|V is equivalent to the condition that there exists v ∈ V whose stabilizer G v is reductive and (2.8) (In general, LHS(2.8) RHS(2.8).) (c) The following example, provided by J. Dadok, shows that condition (2.6) alone is not sufficient. Let g be a simple Lie algebra and G|(g ⊕ g) the action of the adjoint group on the direct sum of 2 copies of g. Let C be the sum of Cartan subalgebras in each copy. Then (2.6) holds, but this linear group is not polar.

Nilpotent elements and polar linear groups.
Let f be a non-zero element of a simple Lie algebra, included in an sl 2 -triple s, and let (1.1) be the corresponding Z-grading of g. Let d be the depth of this grading. Then each g j , |j| d, is a g 0 -module. Since z(s), the centralizer of s in g, is a subalgebra of g 0 , each g j is a z(s)-module. Since z(s) is a reductive subalgebra of g, we obtain a reductive linear Lie algebra z(s)|g j by taking the image of z(s) in End g j . Recall that Z(s) is the centralizer of s in the adjoint group G of g.
Theorem 2.8. All linear groups Z(s)|g j for j = d or d − 1 are polar.
Proof. We do not know a proof of this remarkable fact without the case-wise verification. For j = d it was pointed out in [EJK20] that this fact follows from Tables there for d even and Remark 2.1 for d odd. For j = d − 1, when d is odd, this follows from Table 1 of [DSJKV20], where all these groups are listed, and from Table 1 of [DK85], where all polar linear groups of simple Lie algebras are listed. Finally, for j = d − 1, where d is even, this fact follows from Tables 2, 3 and 4 of this paper. Indeed, looking at Table 1 of [DK85], one can see that, apart from theta groups, all examples that occur in these tables are polar, except, possibly, for the linear reductive groups Spin 7 ⊗ st(SO 2 ) and Spin 9 ⊗ st(SL 2 ). (Hereafter Spin (resp. st) denotes the spinor (resp. standard) representation.) For both of them dim V / /G = 2. It is shown in the examples below that both are polar.
Remark 2.9. We tried to prove polarity of Z(s)|g j for j = d or d − 1 along the following lines.
Let ε = e 2πi j+2 , where j = d or d − 1, and consider the automorphism σ of g, defined by σ| gs = ε s I, s ∈ Z. Then the fixed point set of σ is g 0 and the ε −2 -eigenspace of σ is V := g −2 + g j . Let G 0 be the algebraic subgroup of the adjoint group G, corresponding to the subalgebra g 0 of g. Then we get a theta group G 0 |V . By [DK85], [EJK20, Theorem 6(c)] this representation is polar.
Let SG 0 = G 0 ∩ SL(g −2 ), and consider the linear group SG 0 |g −2 . By Proposition 1.1 from [K80] there exists an SG 0 -invariant polynomial P on g −2 , such that P (a) = 0 for a = 0 if and only if the orbit SG 0 (a) is closed; moreover such an orbit contains a non-zero multiple of f . Since z(s) = g 0 ∩ g f and the group Z(s)|g −2 is orthogonal, we obtain that Z(s) ⊂ SG 0 and Z(s) = (SG 0 ) f . Let sg 0 = g 0 ∩ sl(g −2 ) and consider the slice representation Z(s)|N f , where N f = V /[sg 0 , f ]. Then by the Luna slice theorem [L72] dim(V / /SG 0 ) = dim(g j / /Z(s)) + 1. Therefore, since dim(V / /G 0 ) dim(V / /SG 0 ) + 1, we obtain that dim(V / /G 0 ) dim(g j / /Z(s)). (2.9) Let v ∈ V be such that G 0 (v) is a closed orbit of maximal dimension. Since G 0 |V is polar, is a Cartan subspace. Let π : V → g j be the projection. Since no nonzero G 0 -orbit in g −2 is closed, we conclude that π is injective on C v . Moreover we think that π(C v ) ⊆ C π(v) but we were unable to prove this. But if this is true, then (2.10) Comparing (2.9) and (2.10), we conclude that dim C π(v) dim(g j / /Z(s)). But by (2.3), the LHS cannot be greater than the RHS, hence dim C π(v) = dim(g j / /Z(s)) and the linear group Z(s)|g j is polar.
Example 2.10. G|V = Spin 7 ⊗ st(SO 2 ). This is a direct sum of two irreducible modules V = V + ⊕ V − , which are 8-dimensional spin modules for SO 7 ; SO 2 acts on V ± by multiplication by t ±1 , t ∈ F × . The set of weights of the g-module V is the highest weights of V ± being Λ ± = 1 2 (ε 1 + ε 2 + ε 3 ) ± δ, and lowest weights being −Λ ± . Let v ± be the corresponding highest weight vectors and v ± lowest weight vectors. Let Then, by Proposition 2.1, all vectors from C := span{v 1 , v 2 } are semisimple. It is easy to see that g(C) is the tangent space to the orbit G(v) at a generic point v ∈ C. It follows that C is a Cartan subspace.
(a) All linear groups Z(s)|g j for j even (resp. odd) preserve a symmetric (resp. skew-symmetric) nondegenerate invariant bilinear form. (b) All linear groups Z(s)|g j for j = d or d − 1 are stable, except for all cases when j = d is odd, and also when j = d − 1 is odd and Z(s)|g d−1 either is isomorphic to st(so n ) ⊗ st(sp m ) with n odd, n < m, or has finitely many orbits.
Proof. (a) can be found in [Pa99] or [EJK20]. By (a), the stability follows when j is even from [L72]. When d is even and j = d − 1, the stability is established by a case-wise verification, by checking that [z(s), C] = g j where C is a Cartan subspace (which exists by Theorem 2.8).
Remark 2.13. It follows from Tables 2, 3, 4 that for an odd nilpotent the Dynkin characteristic contains 2 only if Z(s)|g d−1 has finitely many orbits. Moreover, the Dynkin characteristic contains 2 if g j = 0 for some 1 j d − 1; this follows from the observation that when the Dynkin characteristic contains only 0 or 1, the g 0 -module g j is generated by g j 1 . Remark 2.14. For all nilpotent elements of odd depth in Table 5 the linear group Z(s)|g 1 is not polar, except for the last one.

Generalized cyclic elements and integrable triples
In this section g is a semisimple Lie algebra. The following notion generalizes both notions of cyclic and quasi-cyclic elements, introduced in the Introduction, for j = d and j = d − 1 respectively. Definition 3.1. A generalized cyclic element, attached to a nilpotent element f of g is an element of the form f + E, where E ∈ g j with j 1, E = 0, [E, g 2 ] = 0 and the centralizer of E in g 1 is coisotropic with respect to the symplectic form ω, defined by (1.2). Two generalized cyclic elements f + E and f + E are called equivalent if E is proportional to an element from the orbit Z(s)(E).
Recall the notion of an integrable triple, associated to f , which is defined by properties (I1) -(I3) in the Introduction.
is a non-nilpotent generalized cyclic element, associated to f . (b) The element a, defined by (3.1), determines the integrable triple uniquely.
Proof. By the definition, a = (f 1 + E) + f 2 is the Jordan decomposition of a, where f 1 + E and f 2 are its semisimple and nilpotent parts. Since f 1 + E is a non-zero semisimple element, we conclude that a is not nilpotent. By the definition, a is a generalized cyclic element, proving (a).
Due to uniqueness of the Jordan decomposition, a determines f 1 + E and f 2 , hence it determines

Definition 3.3.
A generalized cyclic element f + E is called integrable if it is obtained from an integrable triple as in (3.1). In this case the element E ∈ g j is called integrable for f .
Note that a generalized cyclic element is integrable if and only if the nilpotent part of its Jordan decomposition lies in g −2 , and also that a generalised cyclic element cannot be nilpotent. Also, of course, any semisimple generalized cyclic element is integrable.
Lemma 3.4. If E ∈ g j , j 1, is a nilpotent element of the Z(s)-module g j , then the element f + E is a nilpotent element of g.

Proof.
Clearly the Zariski closure of Z(s)(f + E) contains f . Hence G(f + E) ⊃ Gf 0 since f is a nilpotent element of g.
Proposition 3.5. An integrable generalized cyclic element f + E, associated to a nilpotent element f of depth d, where E ∈ g j , exists only when j = d − 1 or d.
Proof. In most of the cases the center of the subalgebra m := k 2 g k lies in g d−1 +g d . In a few cases this center has non-zero elements E in g d−2 , but it turns out that if the coisotropy condition in (I2) is satisfied, then f + E is nilpotent, thus not integrable. For exceptional g the only f for which the center of m is larger than g d−1 + g d are of type A 2 + A 1 in E 6 , E 7 , E 8 of depth 4, but the elements from m \ (g d−1 + g d ) are not integrable. This is checked by a case-wise verification, with the aid of computer. The proposition is proved by direct computations in the standard representation for classical g. It would be interesting to find a general proof.
The following theorem classifies the non-nilpotent cyclic elements. Proof. It follows from Lemma 3.4 that f + E is nilpotent if E is a nilpotent element of the Z(s)-module g d .
Conversely, suppose E is not a nilpotent element of the Z(s)-module g d . Then Z(s)(E) contains a non-zero semisimple element E 0 of the Z(s)-module g d , and by Theorem 2.5 (a) we may assume that E 0 ∈ C, a Cartan subspace. Hence, we have to prove that f + E 0 is not a nilpotent element of g. For that, by results of [EKV13], it suffices to prove that f s + E 0 is not a nilpotent element of g. By the results of [EJK20], we may assume that f s is an irreducible nilpotent element of g, in which case g d = C, and the set S g (f ) = {E ∈ C | f + E is semisimple} is a complement in C of a union of m(m+1) 2 hyperplanes, where m = dim g d . In the case when m = 1 this means that S g (f ) = C \ {0}, and we are done.
The remaining cases are the following ones from [EJK20, Table 1]: 4 k−1 , m = 2; 7, m = 2; 11, m = 3; 16, m = 2; 17, m = 2; 18, m = 4. For all but the first one, g is an exceptional Lie algebra, and the case can be checked on a computer. In [EJK20], singular sets (complements to S g (f )) for these cases have been described as unions of certain subspaces. Using GAP package SLA, one checks that f + E is not nilpotent for generic elements E in all possible intersections of these subspaces. In the first case g = so 4k and f corresponds to the partition (2k + 1, 2k − 1). This case is treated in Example 3.7 below.
Consider first the case λ 2 = 0, so that E = λE 1 . We then define f s , f n by declaring that f s acts only on the basis elements x i and f n only on the basis elements y j . Thus the corresponding picture is then Then clearly f s + f n = f and f n commutes both with f s and with E. Moreover both f s and f n are homogeneous of degree −2 since they both act on the standard representation lowering degree by 2. Moreover f s + E is semisimple since the characteristic polynomial of its action in the standard representation is t(t 2k − λ). Next consider the case λ 2 = ±λ 1 , so that E = λ(E 1 ± E 2 ). Switch to the following basis: In this basis, the non-zero actions of f and E are as follows: and Et −k = λt k−1 , Ez −(k−1) = λz k . In this basis, let us define f s and f n by the non-zero actions Then again both f s and f n are in g −2 , f s + f n = f , f n commutes with both f s and E, and f s + E is semisimple. Note also that f s + E is not regular, having double eigenvalues. Its characteristic polynomial in the standard representation is (t 2k − λ) 2 (the minimal polynomial being t 2k − λ). All this is completely obvious from the following diagram of actions: Thus in both cases (f 1 , f 2 , E) with f 1 = f s and f 2 = f n are integrable triples.
Example 3.8. According to [EJK20, Table 1], there are three irreducible nilpotent orbits of rank 2 in exceptional Lie algebras: the one with label F 4 (a 2 ) in F 4 and the orbits with labels E 8 (a 5 ) and E 8 (a 6 ) in E 8 . For all of them there are exactly three lines in g d such that f + E is semisimple if and only if E ∈ g d does not belong to any of these lines. Direct calculations with the GAP package SLA show that E remains integrable along these lines too. We provide more detailed description using specific choices of representatives for these orbits as follows.
In F 4 , take for the representative of F 4 (a 2 ) the element In E 8 , take for the representative of E 8 (a 5 ) the element . Then, for these choices of f , g d has basis E 1 , E 2 consisting of the highest root vector and the next to the highest root vector. The cyclic element f + x 1 E 1 + x 2 E 2 is semisimple except when x 1 = 0, x 2 = 0 or x 1 = x 2 . In these three singular cases, Jordan decomposition of f + E is (f s + E) + f n with f n ∈ g −2 , where for E = x 1 E 1 or E = x 2 E 2 , we have the following cases: if f is F 4 (a 2 ), then f s has label B 3 and f n has label A 2 , if f is E 8 (a 5 ), then f s has label E 6 and f n has label D 4 , ), then f s has label C 3 and f n has label A 1 , if f is E 8 (a 5 ), then f s has label D 7 and f n has label 2A 1 , and if f is E 8 (a 6 ), then for E on any of the three singular lines f s has label D 6 and f n has label A 3 .
Note also that the subalgebra a generated by f and E, which is the whole g if E does not lie in the union of singular lines, is the direct sum of the semisimple [a, a] and the center spanned by f n , where [a, a] has the following type: , while if f is E 8 (a 6 ), then for E from any of the three singular lines, this subalgebra has type B 5 . Example 3.9. For the nilpotent with label E 7 (a 5 ) in E 7 , depth is 10, with g 10 3-dimensional. We take . The singular set, i. e. the subset of those E ∈ g 10 with f +E not semisimple, is the union of six 2-dimensional subspaces: Their pairwise intersections produce the following seven 1-dimensional subspaces: It is also possible to describe this set without mentioning semisimplicity of f + E: it is the set of all those vectors which have nontrivial stabilizer with respect to the action Z(s)|g 10 which is the permutation representation of the symmetric group on three letters.
All these subspaces are exactly all those subspaces V of g 10 with the following property: the Lie subalgebra a generated by V and f is the direct sum of the semisimple subalgebra q = [a, a] and the 1-dimensional center z(a) ⊆ g −2 ; z(a) is spanned by an element f n such that f s = f − f n is of semisimple type in q; f s has the same depth 10 in q as f in g; and q 10 = V .
In all these cases, taking any E ∈ q 10 = V which does not lie in a smaller subspace from the singular set, f s + E is semisimple, so that one obtains an integrable triple (f 1 , f 2 , E) = (f s , f n , E).
The corresponding subspaces and nilpotent types are as follows: The singular set in this case is the union of ten 3-dimensional subspaces , forming a single orbit under the action Z(s)|g 10 , which is the action of the component group S 5 on its 4-dimensional irreducible representation.
As in the Example 3.9, this set consists precisely of those vectors which have nontrivial stabilizer with respect to the action of the component group of Z(s)|g 10 .
All of the possible intersections of these 3-dimensional subspaces produce twenty five 2-dimensional subspaces forming two orbits, one containing all E i , E j and all E i , E j +E k +E and another containing all E i , E j + E k and all E i + E j , E k + E , and fifteen 1-dimensional subspaces forming two orbits, one containing all E i and E 1 + E 2 + E 3 + E 4 , and another containing all All these 10 + 25 + 15 = 50 subspaces are also exactly all those subspaces V of g 10 with the following property: the Lie subalgebra a generated by V and f is the direct sum of the semisimple (in fact, simple) subalgebra q = [a, a] and the 1-dimensional center z(a) ⊆ g −2 ; z(a) is spanned by an element f n such that f s = f − f n is of semisimple type (in fact, irreducible) in q; f s has the same depth 10 in q as f in g; and q 10 = V .
In all these cases, taking any E ∈ q 10 = V which does not lie in a smaller subspace from the singular set, f s + E is semisimple, so that one obtains an integrable triple ( The subspaces, their generic stabilizers in S 5 , the corresponding subalgebras and nilpotent orbit labels are as follows: V generic stabilizer The following lemma helps to establish non-existence of non-nilpotent quasi-cyclic elements. Lemma 3.11. Let C ⊂ g d−1 be a Cartan subspace of the Z(s)-module g d−1 , and suppose that C contains no elements E, Proof. If f + E is not nilpotent quasi-cyclic, then, by Lemma 3.4, E is not a nilpotent element for the Z(s)-module g d−1 . Then, as in the proof of Theorem 3.6, In order to prove the next proposition, we will need the following simple lemma.
Lemma 3.12. Let a ∈ g. Then (Ker ad a) ⊥ = Im ad a, where the orthogonal is with respect to the bilinear form (· | ·). In particular, if a ∈ g k , then (Ker ad a| g ) ⊥ = Im ad a| g − −k .
Proof. The inclusion Im ad a ⊂ (Ker ad a) ⊥ is immediate by the invariance of the bilinear form. On the other hand dim(Im ad a) = dim(g) − dim(Ker ad a) = dim (Ker ad a) ⊥ , since the bilinear form is non-degenerate. Hence, (Ker ad a) ⊥ = Im ad a. The second part of the lemma follows from the fact that (g k | g ) = 0 if k = − . Proposition 3.13. Let E ∈ g j , j 0. The following statements are equivalent: (a) the centralizer of E in g 1 is coisotropic with respect to the bilinear form (1.2); Proof. The coisotropy condition (a) can be rephrased by saying that the orthogonal complement of g E 1 with respect to ω is contained in g E 1 . This, by definition (1.2) of ω, is equivalent to the following condition: (Here the orthocomplement is with respect to the bilinear form (· | ·) of g.) On the other hand, the statement in b) can be equivalently rephrased as follows: for some a ∈ g 1 , then a ∈ Ker ad E| g1 , or, equivalently, By Lemma 3.12, (Ker ad E| g1 ) ⊥ = Im ad E| g−j−1 , thus (a') and (b") are equivalent.
An important problem is when a (non-nilpotent) generalized cyclic element f + E of a semisimple Lie algebra g is integrable. For the solution of this problem the following lemma is important.
Lemma 3.14. Let f be a non-zero nilpotent of g and let (1.1) be the corresponding Z-grading of depth d.
The following theorem describes all integrable cyclic elements, up to conjugation by Z(s).
Theorem 3.15. Let f be a non-zero nilpotent element of g of even depth d, and let E ∈ g d be a non-zero element. Then the cyclic element f + E is integrable if and only if E is a semisimple element of the Z(s)-module g d .
Proof. If f + E is integrable, then E is a semisimple element of Z(s)|g d by Lemma 3.14.
Conversely, let E be a non-zero semisimple element of Z(s)|g d . Then the argument as in the proof of Theorem 3.6 reduces the proof to the case when f is an irreducible nilpotent element with dim g d > 1 and E / ∈ S g (f ). Again, the cases when g is an exceptional Lie algebra are checked on the computer using GAP [SLA], by computing Jordan decompositions of f + E for E generic in all possible nonzero intersections of subspaces constituting the complement of S g (f ) as described in [EJK20] (see Examples 3.8, 3.9 and 3.10), while the case g = so 4k , with f corresponding to the partition (2k + 1, 2k − 1) is treated as in Example 3.7.

Theorem 3.16. (a) Theorem 3.15 holds for d odd and for
The "only if" part of Theorem 3.15 holds for d even and for E ∈ g d−1 such that f + E is a quasi-cyclic element.
Proof. (a) Replacing g with g even = j∈Z g 2j , the proof is the same as of Theorem 3.15. (b) follows from Lemma 3.14.
Example 3.17. A non-nilpotent quasi-cyclic element does not necessarily give rise to an integrable triple. This is not an integrable triple because f 2 / ∈ g −2 .
4. Integrable cyclic and quasi-cyclic elements associated to nilpotent elements of even depth 4.1. Integrable cyclic elements for nilpotent elements of even depth. Let f be a non-zero nilpotent element of even depth d in a simple Lie algebra g. Recall that f is included in an sl 2 -triple s, and that, by Theorem 2.8, the linear group Z(s)|g d is polar. All these linear groups are listed in [EKV13]. Let C ⊆ g d be a Cartan subspace. By Theorem 3.15, any cyclic element f + E, where E ∈ C is non-zero, is integrable. Hence, up to conjugation by Z(s), the integrable cyclic elements are classified by non-zero elements of C, up to conjugation by its Weyl group, and rescaling. Recall also by [EKV13] that the set of non-zero nilpotent elements in g (up to conjugation) is partitioned in bushes, such that each bush contains a unique nilpotent element f s of semisimple type, and all other nilpotent elements in the same bush have the same depth d and the same Cartan subspace.
Below we give a more explicit description of Z(s)|g d (rather their unity components) for all classical simple Lie algebras g. As in [DSJKV20], throughout the paper, we use the following notation: st(a) denotes the standard representation of the Lie algebra a, 1 stands for the trivial 1-dimensional representation, ⊕ stands for the direct sum of linear reductive groups, rank = dim g d / /Z(s).
4.1.1. g = sl N , N 2. Non-zero nilpotent elements f , up to conjugation, are parametrized by partitions where the p i are distinct and have multiplicities r i : p 1 > · · · > p s 1, p 1 > 1. Then the associated to a partition p nilpotent element f = f p is of semisimple type if and only if The bush containing this partition consists of all partitions with the same p 1 and r 1 . All these partitions have the same depth d = 2p 1 − 2, and the same g d = Mat r1×r1 , and the action of Z(s)|g d is the action of SL r1 on Mat r1×r1 by conjugation. A Cartan subspace C is the subspace of all diagonal matrices.
4.1.2. g = sp N , N 2 even. Non-zero nilpotent elements f p , up to conjugation, are parametrized by partitions p, whose odd parts have even multiplicity. Then again f = f p is of semisimple type if and only if (4.2) holds. The bush containing f p consists of all partitions (whose odd parts have even multiplicities) with the same p 1 and r 1 as p. All have the same depth d = 2p 1 − 2, and the same linear group Z(s)|g d , which depends on whether p 1 is even or odd: For the bilinear form with matrix I, defining SO r1 , S 2 st(SO r1 ) is identified with the space of all symmetric matrices, and we can choose for C the subspace, consisting of diagonal matrices, while S 2 st(Sp r1 ) is the adjoint representation of Sp r1 , so that C is any Cartan subalgebra. 4.1.3. g = so N , N 3, N = 4. Non-zero nilpotent elements f p , up to conjugation, are parametrized by partitions p, whose even parts have even multiplicity. There are five types of elements f p of semisimple type: The linear groups Z(s)|g d for the types (a) -(e) are as follows: Cartan subspaces are as follows: Bushes containing these f of semisimple type correspond to the following partitions (with all even parts having even multiplicities): (a) partition itself; (b) all partitions with the same p 1 and r 1 = 1 satisfying p 2 < p 1 − 2; (c) all partitions with the same p 1 , r 1 = 1 and p 2 = p 1 − 2; (d) and (e) all partitions with the same p 1 with multiplicity r 1 or r 1 + 1. The group Z(s)|g d is the same for the nilpotent elements from the bush, except for the following two cases: The information about Lie algebra actions of centralizers of sl 2 -triples can be summarized in the following table. We associate to p a symmetric (with respect to the y-axis) pyramid, with boxes of size 2 × 2 indexed by the set I = {1, 2, . . . , N } (say starting from right to left and bottom to top). For example, for the partition (9, 7, 4 (2) ) of 24, we have the pyramid in Figure 1. Let V be the N -dimensional vector space over F with basis {e α } α∈I . The Lie algebra g ∼ = gl(V ) has a basis consisting of the elementary matrices E α,β , α, β ∈ I. The elementary matrix E α,β in g can be depicted by an arrow going from the center of the box β to the center of the box α. In particular, f is the "shift to the left" operator. It is depicted as the sum of all the arrows pointing from each box to the next one on the left where the sum is over all adjacent boxes (on the same row) α, β ∈ I. Let us also denote by f the transpose of the matrix f defined in (4.3). It is the "shift to the right" operator.
Let h ∈ g be the diagonal endomorphism of V whose eigenvalue on e α is the x-coordinate of the center of the box labeled by α (see Figure 1) which we denote by x α . We then have the corresponding We note that the elements f and h belong to a sl 2 -triple where the sum is over all boxes γ at the right and in the same row of the box β, including it.
The elementary matrices E α,β are eigenvectors with respect to the adjoint action of h: This defines a Z-grading of g, given by the ad h-eigenspaces as in (1.1): The depth of this grading is d = 2D = 2p 1 − 2. Next, consider the subspaces V − = Ker f and V + = Ker f of V . We thus have the direct sums decompositions (4.6) Let D i = p i − 1, for i = 1, . . . , s (in particular D 1 = D). Throughout the paper we will use the decompositions Representing the basis elements of V as boxes of the pyramid as in Figure 1, V i corresponds to the i-th rectangle counting from the bottom, and V ±,i correspond to the right/left most boxes of the i-th rectangle. With a picture: For the pyramid in Figure 2 the subspaces V − and V + correspond, respectively, to the boxes colored in orange and blue (note that they may have nontrivial intersection); the subspaces V i , i = 1, 2, 3, correspond to the rectangles of the pyramid, and V ±,i is the intersection of the rectangle V i with V ± .
Throughout the paper, given a subspace U ⊂ V , together with a "natural" splitting V = U ⊕ W (usually associated with the grading of V ), we shall denote, with a slight abuse of notation, by 1 U both the identity map U ∼ −→ U , the inclusion map U → V , and the projection map (with kernel W ) V U ; the correct meaning of 1 U should then be clear from the context.
Using the above notation, we clearly have (recall the splitting (4.6)) (4.9) The following result will be used in the sequel.
Proof. By the direct sum (4.8) we can write 1 Vi x1 (4.10) for every k = 0, . . . , D i − 2. Multiplying both sides of (4.10) on the right by f and using (4.9) we get A recursive solution to these equations is Letting t i = f x 0 ∈ End(V +,i ) we get the claim.
In order to apply, in the following sections, Proposition 3.13 we need the following result. Then Proof. Without loss of generality, let us assume that First, let us assume that i < j. Applying ad f to the RHS of equation (4.11) we get In the first equality we used the first equation in (4.9), while, for the second equality, the operator 1 V+ on the left forces k = k i . Since i < j, by the observation after equation (4.12) we have k i > k j . Hence, xf ki = 0 thus proving (4.11) in this case.
Next, let us assume that i > j and let us apply ad f to the RHS of equation (4.11): In the first equality we used the second equation in (4.9), while, for the second equality, the operator 1 V− on the right forces k = D j − k j . Since i > j, by the observation after equation (4.12) we have Proof. We have that [E, U ] = U A − BU . Since U A ∈ Hom(V 2 , V 1 ) and BU ∈ Hom(V 1 , V 2 ), using Lemma 4.2 we have that The claim follows from the fact that (f ) D : The next result will be used in Sections 4.2.3 and 5.1. (4.13) (4.14) Finally, applying ad E to both side of equation (4.14) we get (4.15) Note that and Hence, the RHS of (4.15) vanishes if and only if equation (4.13) holds.
Finally, the last result of this section will be used in Section 4.5.2..

Lemma 4.5. Let
). Let us assume that Hence, by a straightforward computation we get 4.2.2. The centralizer z(g 2 ). From (4.5) we have that a homogeneous element E ∈ g k , k ∈ Z, has the form The goal of this section is to describe the centralizer z(g 2 ) of g 2 in g.
Proof. Since the adjoint action of e ∈ g 2 is injective on g <0 , we obviously have z(g 2 ) ⊂ g 0 . Letα,β ∈ I be such that xα = D and xβ D − 2 (i. e. the boxβ is completely at the left of the boxα). Then Eαβ ∈ g 2 . Hence, letting E ∈ g k be as in equation (4.17), we have (4.18) If k 1, then the condition x α − D = k 1 implies that x α 1 + D thus the second sum in (4.18) is empty. Hence, from equation (4.18) we have that from which follows that, for x α , x β = D − 1, we have c α,β = δ αβ λ, for some λ ∈ F. Similarly, letting α,β ∈ I be such that xα = −D and xβ −D + 2 (i. e. the boxβ is completely at the right of the box α), the condition [E, Eαβ] = 0 implies, for k 1, that c αβ = 0 if x β −D + 2 and, for k = 0, that c α,β = δ αβ λ for x α , x β = −D + 1. This proves that Proposition 4.7. The centralizer of g 2 in g is z( Proof. By degree consideration and the fact that the identity is a central element we clearly have This, combined to Lemma 4.6, completes the proof. Recalling the definition of integrable triples given in the Introduction, if E is an integrable element for f , then E ∈ (W ⊕ g d−1 ⊕ g d ) ∩ g 1 (note that g d−1 ⊕ g d ⊂ g 1 for p 1 2, and W ⊂ g 1 for p 1 3).
This implies that the minimal polynomial of A dividesq(x). Sinceq(x) has distinct roots, A is semisimple, as claimed.
Proof. Note that f 1 V 2 is nilpotent and commutes with f 1 V1 + E = (f + E)1 V1 . Hence, for the Jordan Since, by assumptions, f + E is integrable, we then have with f 1 +f 2 = f 1 V1 . By Lemma 4.1, we have that and repeating the same computation k times, (4.25) Letting k = D = p 1 − 1 in (4.25), and applying A one more time, we get is nilpotent. Let us also denote by Proposition 4.14.

Similarly, one can check that for every
which is nilpotent and commutes with (f + E)1 U . Hence, the Jordan decomposition of f + E is which omplies that f + E is integrable.
Conversely, if f + E is integrable, in particular g E 1 must be coisotropic. Hence, by Proposition 4.8, ab = λ1 V [D] , for some λ ∈ F. On the other hand, it must be λ = 0 otherwise, as one can easily check, f + E is nilpotent.
The next result will be used in Section 4.5.1. We state and prove it here since we need the notation introduced here.

Assume that f + E is not nilpotent and that its nilpotent part lies in g −2 . Then ab ∈ End(V [D]) is a non-zero semisimple element.
Proof. Note that f 1 V 3 is nilpotent and commutes with f 1 V 2 + E = (f + E)1 V 2 . Hence, for the Jordan By assumption,f 2 = (f 1 V 2 + E) n ∈ g −2 . In particular,f 2 ∈ g f −2 ∩ (End(V 1 ) ⊕ End(V 2 )). Hence, by Lemma 4.1, we have thatf , Repeating the same computation D times, we get (cf. equation (4.25)) A D v = f D v, and applying A one more time we get For the last equality we used (4.28). By equations (4.27) and (4.30) we have where in the last equality we used (4.29). Repeating the same computation D − 1 times we get Applying A again, we finally get Since A is non-zero semisimple, A 2D+1 is non-zero semisimple as well. Moreover, from equation (4.31) = ab thus showing that ab is non-zero semisimple and concluding the proof.

No integrable elements for f in W .
In this subsection we will use the decomposition Proof. By contradiction, let E = c(f ) D−1 ∈ W , where c is as in (4.22), be integrable for f . Since g E 1 is coisotropic, by Proposition 4.9 we have c 2 = 0. Clearly, f + E = (f + E)1 V2 + (f + E)1 V =2 , and f + E preserves the direct sum decomposition (4.32). Note that (f + E)1 V =2 = f 1 V =2 which is nilpotent. On the other hand, it is not difficult to check that if v ∈ V 2 , then we have (f + E) 2D v = c 2 v = 0. Hence, (f + E)1 V2 is nilpotent as well. This proves that f + E is nilpotent, contradicting the fact that it is integrable.
As a consequence of Propositions 4.12, 4.14 and 4.16 we get the following.
Corollary 4.17. If E ∈ g k , k 1, is an integrable element for f ∈ gl N or sl N , then k = d or k = d − 1. Then condition (4.34) on an element from C means that λ 2 1 = ... = λ 2 r1 = λ 2 , hence λ j = ±λ for all j. Since the Weyl group of C contains all sign changes of diagonal elements, up to the action of the Weyl group and rescaling, C contains a unique integrable element (1 r1 0) ⊕ (1 r1 0) . Due to Theorem 3.16 (b), any integrable E ∈ g d−1 is Z(s)-conjugate to C. Thus we obtain the following theorem. Remark 4.20. It follows from [EKV13] and the above discussion that for a nilpotent element f ∈ sl N there exists E ∈ g j , such that the element f + E is semisimple if and only if j = d and p 2 = 1, or j = d − 1 and r 1 = r 2 and p 3 = 1. This claim was stated in [DSKV13], where the associated integrable Hamiltonian systems were also discussed.

General setup for symplectic and orthogonal Lie algebras.
Recall from [CMG93] that nilpotent orbits of sp N (respectively so N ) are in one-to-one correspondence with partitions p of N as in (4.1) with the property that if p a is odd (respectively even), then r a is even, 1 a s.
Let V be the N -dimensional vector space over F with basis {e α } α∈I , where I is an index set for the basis, which can be identified with the set of boxes in the pyramid associated to p (cf. Fig. 1). Given α ∈ I we let α ∈ I correspond to the box in the same rectangle as α reflected with respect to the center of the rectangle. For example, in Fig. 1 Proof. From equations (4.36) and (4.37) we have e β |e α = β δ α ,β = α δ α ,β = η α δ α,β = η e α |e β , α, β ∈ I .
The claim follows.
The following commutation relations hold (α, β, γ, η ∈ I): If we depict, as in Section 4.2.1, the basis elements of V as boxes of a symmetric pyramid associated to the partition p (cf. Fig. 1), let, as usual, f be the endomorphism which corresponds to "shifting to the left". Then, f ∈ g. Indeed, using the second property in (4.36), we have f = α←β E α,β = 1 2 α←β F α,β . Note that the "shift to the right" operator f lies in g as well.
As in Section 4.2.1, let h ∈ End V be the diagonal endomorphism of V whose eigenvalue on e α is the x-coordinate of the center of the box labeled by α (see Figure 1) which we denote by x α . We then have the h-eigenspace decomposition of V (4.4) where D = p 1 − 1 is the maximal eigenvalue of h.
Note that x α = −x α and we have Clearly, the matrices F α,β are eigenvectors with respect to the adjoint action of h: This defines a Z-grading of g, given by the ad h-eigenspaces as in (1.1): Hence, the adjoint ofĀ ∈ End V [D] with respect to β 1 is (4.46) As a consequence, we have a bijection 4.4.1. The centralizer z(g 2 ). From Lemma 4.23 we have that E ∈ g k can be decomposed as Proposition 4.24. The centralizer of g 2 in g is z( Proof. As in Lemma 4.6 we have z(g 2 ) ⊂ g 0 . Letα,β ∈ I be such that xα = D and xβ D − 2 (the boxβ is completely at the left of the boxα). Then Fαβ ∈ g 2 . Hence, letting E as in (4.48), and using the commutation relations (4.41) and equation (4.40) we get If k 1, then the condition x α − D = k 1 is empty. Hence, which implies that that c αβ = 0 if x α D − 2. Using the second equation in (4.48), we have also that c αβ = 0 if x β −D + 2. If k = 0, a similar argument to the one used in the proof of Lemma 4.6 shows that equation (4.49) implies E = 0. As a consequence, E ∈ W ⊕ g d−1 ⊕ g d . On the other hand, by Proposition 4.7, we have Recalling the definition of integrable triples given in the Introduction, by Proposition 4.24, if E is an integrable element for f , then E ∈ (W ⊕ g d−1 ⊕ g d ) ∩ g 1 (note that g d−1 ⊕ g d ⊂ g 1 for p 1 2, and W ⊂ g 1 for p 1 3).

The coisotropy condition. An element E ∈ g d−1 can be uniquely written as
Hence, by (4.47), if p 1 is odd, then a = 0 since it lies in the center of sp(V [D], β 1 ). On the other hand, if p 1 is even, a = 0 since it is a skewadjoint operator commuting with all selfadjoint operators.  Proof. By contradiction, let E ∈ g d−1 be an integrable element for f . Since g E 1 is coisotropic, by Proposition 4.25, we have a = 0. Clearly, f This proves that f + E is nilpotent, contradicting the fact that it is integrable. The proof of the claim for E ∈ W is the same as the proof of Proposition 4.16.
As a consequence of Propositions 4.27 and 4.28 we get the following. 4.5. Integrable triples in g = so N for nilpotent elements of even depth. Recall that for η = 1 in (4.36), the algebra (4.38) is g so N .
Lemma 4.30 (cf. [EKV13]). The depth d of the grading (1.1) for g = so N is We are left to consider the case r 1 = 1 and p 2 p 1 − 2. In this case d = 2D − 2. Indeed, F α1,αp 1 −1 ∈ g 2D−2 , where α 1 is the rightmost box of the pyramid (which is in the bottom row) and α p1−1 is the second leftmost box of the bottom row, is a non-zero element.

Even depth d = 2D.
Proposition 4.31. We have that z(g 2 ) = W ⊕ g d−1 ⊕ g d , where Proof. Similar to the proof of Proposition 4.24.
(i) Let, as in Section 4.4.2, ). If p 1 is even and r 1 = 2, then the subspace g E 1 is coisotropic with respect to the bilinear form (1.2). In the other cases, g E 1 is coisotropic if and only if a = 0. On the other hand, if p 1 is even (this implies r 1 even), then a is a skewadjoint operator (with respect to a skewsymmetric bilinear form) commuting with all selfadjoint operators with respect to the same bilinear form. This gives no condition on a when r 1 = 2, but implies that a = 0 for r 1 > 2. This proves part i). The proof of part ii) is similar to the proof of Proposition 4.9.
Let p 1 be even and where the bilinear form β 1 on V [D] defined in (4.44) is skew-symmetric by (4.45). Then, clearly For every w ∈ V , let us denote by φ(w) : V → F the linear functional φ(w)(w 1 ) = w|w 1 , w 1 ∈ V . Hence, we can write Indeed, using the fact that β 1 (w 1 , w 1 ) = 0, for every w 1 ∈ V [D] we have Hence, in this case, E can be uniquely written as (4.53)

Lemma 4.33. With respect to the basis {u, v} of V [D] we have that
where β 2 is defined by (4.44).
Proof. We have, by (4.51) and (4.52), which gives the first coloumn in the matrix (4.54). Similarly for the second column.
Proposition 4.34. Let E ∈ g d−1 be as in (4.53) and let a ∈ End(V [D]) be as in (4.54). If a is a non-zero semisimple element, then (f + E) n ∈ g −2 .
Proof. By assumption, a is a non-zero semisimple element. In (4.54) a is represented by a 2 × 2 traceless matrix. Hence, it is non-zero semisimple if and only if det(a) = 0. This implies that x and y are linearly Since β 2 is non-degenerate and det(a) = 0 it easily follows that As a consequence, applying f repeatedly, we get Note that (f + E)1 U ⊕V 3 =f 1 U ⊕V 3 is nilpotent and commutes with (f + E)1 V1⊕Vx⊕Vy . Moreover, it is not difficult to check that Let q(t) denote the minimal polynomial of a, which has non-zero distinct roots since a is non-zero semisimple. Equation (4.55) implies that the minimal polynomial of (f + E)1 V1⊕Vx⊕Vy divides q(t 2D+1 ) which obviously has also distinct roots. Then (f + E)1 V1⊕Vx⊕Vy is semisimple. In conclusion, the Jordan The next result characterizes integrable elements for so N when the depth of the grading is d = 2D.
Theorem 4.35. Let g = so N and let f ∈ g be a non-zero nilpotent element of depth d = 2D, D = p 1 − 1.

(i) Let E ∈ g d and let, as in Section 4.2.4, u = Ef D ∈ End(V [D]). Then E is integrable for f if and only if u is semisimple. (ii) Let E ∈ g d−1 and let, as above
. Then E is integrable for f if and only if p 1 is even, r 1 = 2 and det a = 0. (iii) If E ∈ W , then E cannot be integrable for f . Proof. Part i) follows by Proposition 4.12. If p 1 is even and r 1 = 2 part ii) follows from Proposition 4.32i), Lemma 4.15 and Proposition 4.34. The remaining claim in part ii) and iii) can be proved in the same way as for the proof of Proposition 4.28 using Proposition 4.32.
Remark 4.36. We have a non-zero E ∈ g d−1 if and only if p 2 = p 1 − 1 (otherwise dim g d−1 = 0). In this case, if r 2 = 1, there are no elements E ∈ g d−1 satisfying the assumption of Theorem 4.35ii). Indeed, since dim V [D − 1] = r 2 = 1, x and y are linearly dependent thus a defined in (4.54) is nilpotent.

Even depth
Proposition 4.37. The centralizer of g 2 in g is z(g 2 ) = g d−1 ⊕ g d .
Proof. Clearly, z(g 2 ) ⊂ g 0 . By degree considerations, z(g 2 ) ⊃ g d−1 ⊕ g d . On the other hand, let 1 ∈ I be the label of the rightmost box of the pyramid associated to p (note that x 1 = D) and p 1 ∈ I be the label of the leftmost box (note that x p1 = −D). Let alsoβ ∈ I be such that xβ D − 2 (the boxβ is completely at the left of the box 1). Then F 1β ∈ g 2 . Hence, letting E as in equation (4.48), using the commutation relations (4.41), the second equation in (4.48) and (4.40), we have that In the sequel, we are going to use the following basis of V 1 . Let v + ∈ V +,1 be such that β 1 (v + , v + ) = 1. Then we consider the basis (4.58) for some λ ∈ F and a ∈ V +,2 .
Let now E ∈ g d−1 instead. It can be uniquely written as . Hence, B = bφ(v + ), for some b ∈ V +,3 , and we can write (4.59) We also set (4.60) Proposition 4.38. Let E ∈ g d−1 be as in (4.59) and let β be as in (4.60). The subspace g E 1 is coisotropic with respect to the bilinear form (1.2) if and only if β = 0.
Since X is arbitrary we get β = 0.
The next result characterizes integrable elements for so N when the depth of the grading is d = 2D − 2.  Proof. In order to prove part (i) one can use the same arguments that will be used for the proof Theorem 5.6 (ii) (in fact, part (i) of the present Theorem corresponds to the special case b = 0 of Theorem 5.6 (ii)). Let us then prove part (ii). Let E ∈ g d−1 be as in (4.59) and let β be as in (4.60). If E is integrable for f , then g E 1 must be coisotropic. By Proposition 4.38, then β = 0, and, as one can easily check, f + E is nilpotent. This contradicts the fact that E is integrable and proves part (ii).

Let us reformulate the results, obtained in Subsection 4.5 about integrable quasicyclic elements in terms of polar linear groups.
First, actions of centralizers of the sl 2 -triples for nilpotent elements in simple Lie algebras of classical types on g d−1 are given in the following table: Table 2. Actions of centralizers of the sl 2 -triples for nilpotent elements in simple Lie algebras of classical types on g d−1 In Tables 2 and 3, for a G-module V , D(V ) stands for the G-module V ⊕ V * .
Cartan subspaces for the entries in Table 2 are as follows.
Proof. (a) follows from Table 2, since all these linear groups are theta groups. In order to prove (b), note that in the case in question, Z(s)|g d−1 = st(Sp 2 ) ⊗ st(SO r2 ), for which the rank equals 1. This shows that, up to equivalence, there is at most one integrable quasi-cyclic element. Its existence follows from Theorem 4.35 (ii). 4.6. Integrable quasi-cyclic elements in exceptional Lie algebras for nilpotent elements of even depth.  Table 3. Quasi-cyclic elements, attached to odd nilpotent elements f of even depth in exceptional g with Z(s)|g d−1 having non-trivial invariants By the quasi type in the last column we mean whether f + E for a generic E ∈ g d−1 is semisimple or mixed (meaning neither semisimple nor nilpotent).  Table 4. Odd nilpotent elements f of even depth in exceptional g, for which Z(s)|g d−1 has only trivial invariants 4.6.1. Nilpotent elements with label A 1 . A representative for the orbit with label A 1 is given by a negative root vector e −α -arbitrary for type E and a long one for F 4 and G 2 . Depth is 2, and a 1-dimensional Cartan subspace for z(s)|g 1 is spanned by the vector E = v * + v * , where v * , resp. v * is the highest, resp. lowest weight vector. Moreover E satisfies the coisotropy condition, and the quasi-cyclic element f + E is semisimple. 4.6.2. Nilpotent element with label A 1 in F 4 . A representative f for A 1 is given by a short root vector. Depth is 2, with g 1 of dimension 8. The representation z(s)|g 1 is the direct sum V = V 1 ⊕ V 2 of two standard representations of sl 4 , and a Cartan subspace is spanned by E = v 1 + v 2 , where v 1 is a highest weight vector for V 1 and v 2 a lowest weight vector for V 2 . This E does not satisfy the coisotropy condition, which implies that all quasi-cyclic elements are nilpotent. 4.6.3. Nilpotent element with label 2A 1 in E 6 . A representative f for 2A 1 is given by the sum of any two commuting root vectors. Depth is 2, and z(s)|g 1 is as in Example 2.10, so we can choose a basis {E 1 , E 2 } of a Cartan subspace as there, with E 1 = v + + v − and E 2 = v − + v + . Coisotropy condition for E = x 1 E 1 + x 2 E 2 turns out to be x 2 1 = x 2 2 . We then check that both f + x(E 1 + E 2 ) and f + x(E 1 − E 2 ) are semisimple for x = 0. These are all integrable quasi-cyclic elements f + E for E from the Cartan subspace.
In fact these two solutions E 1 + E 2 and E 1 − E 2 are equivalent under the action of Z(s). This can be seen as follows. Take  Hence in the corresponding Z/2Z-grading g = g 0 ⊕g 1 , where g 0 = g 0 and g 1 = g 2 ⊕g −2 , one has f, E 1 ∈ g 0 and E 2 ∈ g 1 . Let α H be the inner automorphism corresponding to this Z/2Z-grading, i. e. α H (x) = x for x ∈ g 0 and α H ( Coisotropy condition for E = x 1 v 1 + x 2 v 2 turns out to be x 2 1 + x 2 2 = 0 and x 2 1 − x 2 2 = 0, which implies that all quasi-cyclic elements are nilpotent. 4.6.5. Nilpotent element with label 2A 1 in E 8 . As in two previous cases, a representative f for 2A 1 is given by the sum of any two commuting root vectors. Depth is 2, and z(s)|g 1 is spin 13 , so, as in [GV78, Proposition 10], we can choose a basis {v 1 , v 2 } of a Cartan subspace as there, with v 1 = v Λ +v −Λ and v 2 = e −ε1 e −ε2 e −ε3 v Λ + e ε1 e ε2 e ε3 v −Λ , where Λ is the highest weight. Coisotropy condition for E = x 1 v 1 + x 2 v 2 turns out to be, as in the previous case, x 2 1 + x 2 2 = 0 and x 2 1 − x 2 2 = 0, which implies that all quasi-cyclic elements are nilpotent. 4.6.6. Nilpotent elements with label A 2 + A 1 in E 6 , E 7 and E 8 . Here depth is 4, the algebra z(s) is gl 3 for E 6 , gl 2 for E 7 and sl 6 for E 8 . The representation z(s)|g 3 is a direct sum V 1 ⊕ V 2 of two copies of a standard representation of sl 3 for E 6 , of sl 2 for E 7 and of sl 6 for E 8 . A Cartan subspace of g 3 is spanned by where v 1 is a highest weight vector for V 1 and v 2 is a lowest weight vector for V 2 . Each of these E satisfies the coisotropy condition, and the quasi-cyclic element f + E is semisimple. 4.6.7. Nilpotent element with label A 2 + 2A 1 in E 6 , E 7 and E 8 . Depth is 4. Here z(s)|g 3 is V ⊗ st(sl 2 ), where V is st(so 2 ) for E 6 , st(so 4 ) for E 7 and spin 7 for E 8 . In all three cases a Cartan subspace is spanned by where v * denote highest weight vectors and v * the lowest weight vectors, both for V and for st(sl 2 ). This E does not satisfy the coisotropy condition, so that all quasi-cyclic elements are nilpotent. 4.6.8. Nilpotent elements with label A 4 + A 1 in E 6 , E 7 and E 8 . Depth is 4. The representation z(s)|g 3 is 2-dimensional, it is st(so 2 ) with a 1-dimensional Cartan subspace spanned by E = v * + v * , where v * , resp. v * is the highest, resp. lowest weight vector. This E satisfies the coisotropy condition, and f + E is semisimple. 4.6.9. Nilpotent elements with label A 3 + A 2 in E 7 and E 8 . Depth is 6. The algebra z(s) is sl 2 plus a 1-torus for E 7 and sp 4 plus a 1-torus for E 8 . The representation z(s)|g 5 is direct sum V + ⊕V − of two copies of a standard representation, with the torus acting as ±1 on V ± . It has a 1-dimensional Cartan subspace spanned by E = v + + v − , the sum of the highest weight vector of V + and the lowest weight vector of V − . This E does not satisfy the coisotropy condition, so that all quasi-cyclic elements are nilpotent. 4.6.10. Nilpotent element with label A 2 + 3A 1 in E 8 . Depth is 4, and z(s)|g 3 is st(G 2 ) ⊗ st(sl 2 ), with 1-dimensional Cartan subspace spanned by E = v * + v * , the sum of the highest and the lowest weight vectors. This E does not satisfy the coisotropy condition, which means that all quasi-cyclic elements are nilpotent. 4.6.11. Nilpotent element with label A 3 + A 2 + A 1 in E 8 . Depth is 6, and z(s)|g 5 is st(sl 2 ) ⊗ st(so 3 ), with 1-dimensional Cartan subspace spanned by E = v * + v * , the sum of the highest and the lowest weight vectors. This E does not satisfy the coisotropy condition, which means that all quasi-cyclic elements are nilpotent. 4.6.12. Nilpotent element with label A 4 + 2A 1 in E 8 . Depth is 8, the algebra z(s) is sl 2 plus a 1-torus, and the representation z(s)|g 7 is the direct sum V + ⊕ V − of two copies of st(sl 2 ), with the torus acting by ±1 on V ± . It has a 1-dimensional Cartan subspace spanned by E = v + + v − , the sum of the highest and the lowest weight vectors of V + , resp. V − . This E satisfies the coisotropy condition, and the quasi-cyclic element f + E has Jordan decomposition (f s + E) + f n where f s , f n ∈ g −2 are nilpotent elements with labels A 4 + A 1 and A 1 respectively. This gives an integrable triple for this case. 4.6.13. Nilpotent element with label A 4 +A 2 +A 1 in E 8 . Depth is 8, the algebra z(s) is sl 2 , and z(s)|g 7 is its 4-dimensional irreducible representation. It has a 1-dimensional Cartan subspace spanned by E = v * +v * , the sum of the highest and the lowest weight vectors. This E does not satisfy the coisotropy condition, so that all quasi-cyclic elements are nilpotent. 4.6.14. Nilpotent element with label D 7 (a 2 ) in E 8 . Depth is 14, and z(s)|g 13 is the standard representation of so 2 . It has a 1-dimensional Cartan subspace spanned by E = v * + v * , the sum of the highest and the lowest weight vectors. This E satisfies the coisotropy condition, and the quasi-cyclic element f + E is semisimple.
Remark 4.42. It was proved in [DSKV13] that for a long root vector f ∈ g = sp N there exists a unique, up to equivalence, integrable quasi-cyclic element. This covers f of type A 1 in all exceptional g (see Table  3).
Conclusion. Due to Theorem 3.16 (b), Subsections 4.6.1, 4.6.3, 4.6.8, 4.6.12, and 4.6.14 describe all integrable quasi-cyclic elements f + E for nilpotent elements f of even depth for all examples from Table  3, up to conjugation by Z(s). Obviously even nilpotent elements and the nilpotent elements from Table  4 have no integrable quasi-cyclic elements. As a result, we see that for each nilpotent element f of even depth in an exceptional simple Lie algebra either there are no integrable quasi-cyclic elements f + E, or, up to equivalence, there is exactly one.

Integrable quasi-cyclic elements associated to nilpotent elements of odd depth
Recall that if f ∈ g is a nilpotent element of odd depth d, then all elements of the Z(s)-module g d are nilpotent [EKV13, Theorem 1]. Actually the linear group Z(s)|g d is the full symplectic group, [EJK20, Remark 2], hence it is polar. Thus, by Lemma 3.14, if f has odd depth, there are no integrable cyclic elements f + E, with E ∈ g d .
By [EKV13], if g is a classical Lie algebra, nilpotent elements f of odd depth exist only in so n , and for g exceptional, such f are listed in [EKV13, Table 1]. These two cases are treated in Subsections 5.1 and 5.2 respectively. 5.1. Integrable quasi-cyclic elements in so N for nilpotent elements of odd depth. Let g = so N and let f ∈ g be a nilpotent element associated to the partition p as in (4.1). Assume that it has odd depth d = 2D − 1, D = p 1 − 1. By Lemma 4.30, this happens when p 1 is odd, r 1 = 1 and p 2 = p 1 − 1.
We realize the Lie algebra g ad in (4.38), with η = 1, and we let {F α,β , α, β ∈ I} be the set of generators of g defined in (4.39). The first result describes z(g 2 ), the centralizer of g 2 in g.
Proof. The proof is similar to the proof of Proposition 4.37. For completeness, we replicate the argument. Clearly, z(g 2 ) ⊂ g 0 . By degree considerations, z(g 2 ) ⊃ g d−1 ⊕ g d . On the other hand, let 1 ∈ I be the label of the rightmost box of the pyramid associated to p (note that x 1 = D) and p 1 ∈ I be the label of the leftmost box (note that x p1 = −D). Let alsoβ ∈ I be such that xβ D − 2 (the boxβ is completely at the left of the box 1). Then F 1β ∈ g 2 . Hence, letting E as in equation (4.48), using the commutation relations (4.41), the second equation in (4.48) and (4.40), we have that (5.1) If k 1, then the condition x α − D = k 1 implies that x α 1 + D, which is empty. Hence, c αβ = 0 if x α D − 2 and β = p 1 (since F 1,p1 = 0 by (4.40) and (4.36)). Using the second equation in (4.48), we have also that c αβ = 0 if α = 1 and x β −D + 2. Hence, we can write Note that the first sum lies in g d−1 . Let then assume that E = x β =D−k c 1β F 1β and thatα,β ∈ I are such that xα − xβ 2. Then By the discussion at the beginning of this section and Proposition 5.1, if (f 1 , f 2 , E) is an integrable triple, then E ∈ g d−1 .
Recall that dim V [D] = r 1 = 1, and let v + ∈ V [D] be such that f D v + |v + = 1. We consider the basis {f k v + } D k=0 of V 1 (cf. Section 4.5.2). Let also v − = f D v + . Lemma 5.2. There is a bijective correspondence between the triples (λ, a, b), where λ ∈ F, a ∈ V +,3 and b ∈ End(V +,2 ) selfadjoint with respect to the bilinear form β 2 on V +,2 defined by (4.44), and the elements E ∈ g d−1 , given by Proof. An element E ∈ g d−1 can be uniquely written as Furthermore, let us define b = Bf D−1 ∈ End(V +,2 ). Since D 2 = p 2 − 1 = p 1 − 2 = D − 1 and p 1 is odd, then the bilinear form (4.44) β 2 on V +,2 is skewsymmetric. Note that, for v, w ∈ V +,2 we have Hence, b is self-adjoint with respect to β 2 . A similar computation shows that, if b is selfadjoint with respect to β 2 , then B = b(f ) D−1 ∈ g d−1 .
In order to prove the main result of this section we need the following.
Proof. Since α = 0, we have a = 0 and the direct sum decomposition Let V a = ⊕ D−2 k=0 Ff k a and U = ⊕ D−2 k=0 Ff k Ker φ(a)1 V+,3 . Then, we have the direct sum decomposition In this case the Jordan decomposition of X = X s + X n has X s = X1 V1⊕Va and X n = X1 U ∈ g −2 , as claimed. If instead α = λ 2 , then the nilpotent part of X1 V1⊕Va is the following element of g −2 (The proof of this fact is straightforward and is omitted).
The following main result of this subsection characterizes integrable quasi-cyclic elements for so N associated to nilpotent elements of odd depth.
Theorem 5.6. Let g = so N and let f ∈ g be a nilpotent element of odd depth d = 2D − 1, where D = p 1 − 1. Let E ∈ g d−1 be decomposed as in (5.3) and let α be as in (5.8). Then E is integrable for f if and only if the following two conditions hold: (i) b is semisimple with minimal polynomial dividing commutes with both (f + E)1 V2 and (f + E)1 V1⊕V3 , we are left to understand when X = (f + E)1 V1⊕V3 is integrable. By Lemma, 5.4 X is not integrable if a = 0 and α = 0. If a = α = 0, then λ = 0, otherwise E = 0 (indeed if λ = α = 0, then b = 0). When λ = 0, then X is semisimple since its minimal polynomial is x(x D − 2λ) which has distinct roots (see Example 2.12 in [DSJKV20]). If α = 0, then the result follows from Proposition 5.3 and Lemma 5.5.
Remark 5.7. The integrable element E ∈ g d−1 constructed in Example 2.12 in [DSJKV20] corresponds to the choice λ = 1 and a = b = 0 in Theorem 5.6.
Let us reformulate the results obtained in this subsection in terms of polar linear groups.
This proves (a). Claim (b) follows from Theorem 5.8 and claim (a).

Integrable quasi-cyclic elements in exceptional Lie algebras for nilpotent elements of odd depth.
We begin by explaining details of calculations that were used to produce Table 1 in [DSJKV20] (see Table 5 below).  Table 5. Quasi-cyclic elements attached to nilpotent elements of odd depth in exceptional simple Lie algebras As in [DSJKV20], st(a) denotes the standard representation of the Lie algebra a (which is the 26dimensional for a = F 4 ). In this Table rank = dim g d−1 / /Z(s). We call f to be of semisimple (resp. mixed) quasi type if there exist E ∈ g d−1 , such that f + E is semisimple (resp. not nilpotent).
Notation for the nilpotent elements describes them as principal nilpotent elements in the corresponding Levi subalgebras. For example, in E 8 there is a unique, up to conjugacy, Levi subalgebra of type 2A 2 +A 1 ; then f is the sum of the corresponding negative simple root vectors. In E 7 there are two, up to conjugacy, Levi subalgebras of type 3A 1 ; 3A 1 stands for the one whose principal nilpotent has odd depth in E 7 (the principal nilpotent for the other one has even depth). Finally, in F 4 and G 2 tilde means that we take the negative short simple root vector.
Except for the nilpotent with label A 1 + A 1 in F 4 , Cartan subspaces in g 1 with respect to the z(s)-module structure are given by the zero weight spaces of these modules. 5.2.1. Nilpotent elements with label 3A 1 in E 6 , E 8 and 3A 1 in E 7 . All these conjugacy classes have representatives of the form f = e −α1 + e −α2 + e −α3 , sums of three pairwise commuting negative simple root vectors. For E 6 and E 8 these can be arbitrary three commuting root vectors, while in E 7 arbitrary under the restriction that f has odd depth. One checks that in this case the subspace C ⊆ g 2 spanned by e α1 , e α2 , e α3 is a Cartan subspace with respect to the action of Z(s). The coisotropy condition on E = x 1 e α1 + x 2 e α2 + x 3 e α3 is x 2 1 + x 2 2 + x 2 3 − 2x 1 x 2 − 2x 1 x 3 − 2x 2 x 3 = 0.
The subalgebra generated by e ±αi , i = 1, 2, 3, is the direct sum of three copies of sl 2 , and it is straightforward to check that f + E is semisimple when x 1 , x 2 , x 3 are arbitrary nonzero numbers satisfying the coisotropy condition. When one of them is zero, say, x 1 = 0, then the coisotropy condition forces x 2 = x 3 = x = 0, in which case the Jordan decomposition of f +x(e α2 +e α3 ) is (e −α2 +e −α3 +x(e α2 +e α3 ))+e −α1 and we get an integrable triple (f 1 , f 2 , E), where f 1 has label 2A 1 and f 2 has label A 1 .
5.2.2. Nilpotent elements with label 2A 2 + A 1 in E 6 , E 7 and E 8 . This case is described in [DSJKV20, Example 2.14]. Here d = 5. We can take f = e −α1 + e −α2 + e −β1 + e −β2 + e −γ , where α 1 + α 2 and β 1 + β 2 are roots, while no other pairwise sum of the α i , β j and γ is a root. One then checks that the subspace of g d−1 , spanned by e α1+α2 and e β1+β2 , is a Cartan subspace. The coisotropy condition for E = xe α1+α2 + ye β1+β2 is then x = y, and for E = x(e α1+α2 + e β1+β2 ) the Jordan decomposition of the quasi-cyclic element f + E is (e −α1 + e −α2 + e −β1 + e −β2 + x(e α1+α2 + e β1+β2 )) + e −γ . This is straightforward to check after restricting considerations to the subalgebra of type A 2 + A 2 + A 1 containing both f and E. We thus obtain integrable triple (f 1 , f 2 , E), where f 1 has label 2A 2 and f 2 has label A 1 . Clearly in such way we obtain all possible integrable triples that might occur in this case: if all three of the x 1 , x 2 , x 3 are nonzero, we obtain the integrable triple (f, 0, E), while if some of them are zero, we necessarily get the integrable triples (f 1 , f 2 , E) as above. 5.2.3. Nilpotent elements with label 4A 1 in E 7 and E 8 . These conjugacy classes have representatives of the form f = e −α1 + e −α2 + e −α3 + e −α4 , sums of four pairwise commuting root vectors. One checks that in this case the subspace C ⊆ g 2 spanned by e α1 , e α2 , e α3 , e α4 is a Cartan subspace with respect to the action of Z(s). The coisotropy condition on E = x 1 e α1 + x 2 e α2 + x 3 e α3 + x 4 e α4 is where {i, j, k} is any three-element subset of {1, 2, 3, 4} for E 8 , while for E 7 it can be any three-element subset except one of them.
For E 8 the resulting system of quadratic equations has only zero solution, which means, by Lemma 3.11, that in this case there are no non-nilpotent quasi-cyclic elements.
The subalgebra generated by e ±αi , i = 1, 2, 3, 4, is a direct sum of 4 copies of sl 2 , which easily implies that the last four solutions give semisimple quasi-cyclic elements, while the first solution gives a quasi-cyclic element with the Jordan decomposition (e −α2 + e −α3 + e −α4 + x(e α2 + e α3 + e α4 )) + e −α1 , which gives an integrable triple (f 1 , f 2 , E) where f 1 has label 3A 1 and f 2 has label A 1 .

Nilpotent element with label
Here z(s)|g d−1 is the direct sum of a 5-dimensional irreducible and 1-dimensional trivial representation of sl 2 . The subspace of g d−1 spanned by E 0 = e 1220 , E 1 = e 1222 +e 1232 +e 1242 and E 2 = e 1222 −e 1232 +e 1242 is a Cartan subspace. Coisotropy condition on E = x 0 E 0 + x 1 E 1 + x 2 E 2 is x 2 0 + 4x 2 1 + 4x 2 2 − 4x 0 x 1 + 4x 0 x 2 + 8x 1 x 2 = 0. Subalgebra generated by f and the Cartan subspace is a direct sum of three copies of sl 2 , and the matrix of f + E in the standard representation of this subalgebra is It follows that the quasi-cyclic element f + E with E as above satisfying the coisotropy condition, is semisimple except for the cases x 0 = 0, x 1 = −x 2 ; x 1 = 0, x 0 = −2x 2 ; x 2 = 0, x 0 = 2x 1 .
The Jordan decomposition of f + E in these cases is, respectively, In all three cases we thus get an integrable triple (f 1 , f 2 , E), where f 1 has label A 1 and f 2 has label A 1 . These three cases, together with the case when the quasi-cyclic element is semisimple, give all possible integrable triples for this case. 5.2.9. Nilpotent element with label A 2 + A 1 in F 4 . A Cartan subspace is given by the zero weight space of the adjoint representation of sl 2 , and none of its nonzero vectors satisfies the coisotropy condition. It follows from Lemma 3.11 that all quasi-cyclic elements are nilpotent. 5.2.10. Nilpotent element with label A 1 in G 2 . This nilpotent does not produce any quasi-cyclic elements, as Example 2.8 in [DSJKV20] shows. Namely, the depth is 3, and g 2 is 1-dimensional, spanned by e 12 ; its centralizer has zero intersection with g 1 , and the zero subspace is not coisotropic.
Conclusion. Due to Theorem 3.16 (a), Subsections 5.2.1, 5.2.2, 5.2.3, and 5.2.8 describe all integrable quasi-cyclic elements f + E for nilpotent elements f of odd depth, for all exceptional simple Lie algebras, up to conjugation by Z(s). In particular, an integrable quasi-cyclic element exists for such f , except for seven cases, described in the Introduction.