Homogeneous non-degenerate $3$-$(\alpha,\delta)$-Sasaki manifolds and submersions over quaternionic K\"ahler spaces

We show that every $3$-$(\alpha,\delta)$-Sasaki manifold of dimension $4n + 3$ admits a locally defined Riemannian submersion over a quaternionic K\"ahler manifold of scalar curvature $16n(n+2)\alpha\delta$. In the non-degenerate case ($\delta\neq 0$) we describe all homogeneous $3$-$(\alpha,\delta)$-Sasaki manifolds fibering over symmetric Wolf spaces (case $\alpha\delta>0$) and over their the noncompact dual symmetric spaces (case $\alpha\delta<0$). If $\alpha\delta>0$, this yields a complete classification of homogeneous $3$-$(\alpha,\delta)$-Sasaki manifolds; for $\alpha\delta<0$, we provide a general construction of homogeneous $3$-$(\alpha,\delta)$-Sasaki manifolds fibering over nonsymmetric Alekseevsky spaces, the lowest possible dimension of such a manifold being $19$.

1 Introduction and basic notions 1.1 Introduction Sasaki manifolds have been studied since the 1970s as an odd dimensional counterpart to Kähler geometry. Similarly, 3-Sasaki manifolds are considered the (4n+ 3)-dimensional analogue to hyper-Kähler (hK) geometry. However, while these geometries are linked via the hK cone of a 3-Sasaki manifold, 3-Sasaki geometry also connects to another 4n-dimensional geometry, namely quaternionic Kähler (qK) manifolds. Initially shown in the regular case by Ishihara and in full generality by C. Boyer, K. Galicki and B. Mann in '94, every 3-Sasaki manifold locally admits a fibration over a qK orbifold [BGM94]. This led to the classification of all homogeneous 3-Sasaki manifolds. The reverse construction is given by taking the Konishi bundle of a positive scalar curvature qK space, i.e. the orthonormal frame bundle of the quaternionic structure [Ko75]. For qK manifolds with negative scalar curvature one does not obtain a 3-Sasaki manifold but a so-called pseudo 3-Sasaki structure [Ta96]. This notion, however, did not gather as much traction since it comes with a metric of semi-Riemannian signature (4n, 3).
More recently the first two authors investigated Riemannian almost 3-contact metric manifolds by means of connections with torsion [AD20]. They found necessary and sufficient conditions for the existence of compatible connections. Along their investigations, they discovered the more specific class of 3-(α, δ)-Sasaki manifolds connecting many examples on which partial results were known previously. In particular, they showed that pseudo 3-Sasaki structures can be turned into negative 3-(α, δ)-Sasaki manifolds (i. e. with αδ < 0). This paper aims to connect both worlds and presents 3-(α, δ)-Sasaki geometry as the go-to structure above any qK space. We quickly review all necessary notions involving 3-(α, δ)-Sasaki structures in Section 1. Using results by R. Cleyton, A. Moroianu and U. Semmelmann [CMS18] we obtain a locally defined Riemannian submersion over a qK space establishing the canonical connection as the link between both geometries. This is done in Section 2. In the 3-Sasaki case we recover the result of Boyer, Galicki, Mann. We further show that the scalar curvature on the base is a positive multiple of αδ. Thus, for negative and degenerate 3-(α, δ)-Sasaki manifolds we obtain submersions onto qK spaces of negative scalar curvature, respectively hK spaces. This suggests to investigate non-degenerate homogeneous 3-(α, δ)-Sasaki manifolds by looking at homogeneous qK manifolds of non-vanishing scalar curvature. Section 3 is therefore devoted to a hands on construction of homogeneous 3-(α, δ)-Sasaki spaces over all known homogeneous qK manifolds. This yields a construction over symmetric Wolf spaces, deforming the description given in [DOP18] (see also [Bi96,Theorem 4]), and by similar means their non-compact duals. Additionally, homogeneous 3-(α, δ)-Sasaki manifolds over Alekseevsky spaces are constructed using a description of the latter given by V. Cortés in [Co00]. We provide detailed descriptions of the 7-dimensional Aloff-Wallach space, its negative counterpart fibering over the 4-dimensional Wolf space SU(3)/S(U(2) × U(1)), respectively its non-compact dual, as well as of the homogeneous 3-(α, δ)-Sasaki spaceT (1) in dimension 19 sitting above the non-symmetric Alekseevsky space T (1). In Section 4 we compute the Nomizu map associated to the canonical connection, the necessary tool for any further investigation of these spaces. In the symmetric base case we find the Nomizu map of the Levi-Civita connection as well.

Review of 3-(α, δ)-Sasaki manifolds and their basic properties
We review some basic definitions and properties on almost contact metric manifolds. This serves mainly as a reference.
It follows that ϕ has rank 2n and the tangent bundle of M splits as T M = H ⊕ ξ , where H is the 2n-dimensional distribution defined by H = Im(ϕ) = ker η = ξ ⊥ . In particular, η = g(·, ξ). The vector field ξ is called the characteristic or Reeb vector field. The almost contact metric structure is said to be normal if N ϕ := [ϕ, ϕ] + dη ⊗ ξ vanishes, where [ϕ, ϕ] is the Nijenhuis torsion of ϕ [Bl10].
An α-Sasaki manifold is defined as a normal almost contact metric manifold such that dη = 2αΦ, α ∈ R * , where Φ is the fundamental 2-form defined by Φ(X, Y ) = g(X, ϕY ). For α = 1, this is a Sasaki manifold. The 1-form η of an α-Sasaki structure is a contact form, in the sense that η ∧ (dη) n = 0 everywhere on M . The Reeb vector field is always Killing.
An almost 3-contact metric manifold is a differentiable manifold M of dimension 4n+3 endowed with three almost contact metric structures (ϕ i , ξ i , η i , g), i = 1, 2, 3, sharing the same Riemannian metric g, and satisfying the following compatibility relations for any even permutation (ijk) of (123) [Bl10]. The tangent bundle of M splits into the orthogonal sum T M = H ⊕ V, where H and V are respectively the horizontal and the vertical distribution, defined by In particular H has rank 4n and the three Reeb vector fields ξ 1 , ξ 2 , ξ 3 are orthonormal. The manifold is said to be hypernormal if each almost contact metric structure (ϕ i , ξ i , η i , g) is normal. We denote an almost 3-contact metric manifold by (M, ϕ i , ξ i , η i , g), understanding that the index is running from 1 to 3. One of the most interesting classes of almost 3-contact metric manifolds is given by 3-α-Sasaki manifolds, for which each of the three structures is α-Sasaki. For α = 1, this is just the definition of a 3-Sasaki manifold. As a comprehensive introduction to Sasaki and 3-Sasaki geometry, we refer to [BG08]. In the recent paper [AD20] the new class of 3-(α, δ)-Sasaki manifolds was introduced, generalizing 3-α-Sasaki manifolds.
We recall some basic properties of 3-(α, δ)-Sasaki manifolds whose proofs can be found in [AD20]. Any 3-(α, δ)-Sasaki manifold is shown to be hypernormal, thus generalizing Kashiwada's theorem [Ka01]. Hence, for α = δ one has a 3-α-Sasaki manifold. Each Reeb vector field ξ i is Killing and it is an infinitesimal automorphism of the horizontal distribution H, i.e. dη i (X, ξ j ) = 0 for every X ∈ H and i, j = 1, 2, 3. The vertical distribution V is integrable with totally geodesic leaves. In particular, the commutators of the Reeb vector fields are purely vertical and for every even permutation (ijk) of (123) they are given by Meanwhile, the vertical part of commutators of horizontal vector fields is encoded by the fundamental form, as is shown in the following useful lemma: Proof. Since the vertical distribution is spanned by the Reeb vector fields, we have By the same argument [X, Y ] V = 0 if X ∈ H and Y = ξ j , j = 1, 2, 3, which is equivalent to the fact that dη i (X, ξ j ) = 0, i = 1, 2, 3.
A remarkable property of 3-(α, δ)-Sasaki manifolds is that they are canonical almost 3-contact metric manifolds, in the sense of [AD20], which is equivalent to the existence of a canonical connection.
We recall here some basic facts about connections with totally skew-symmetric torsion-we refer to [Ag06] for further details. A metric connection ∇ with torsion T on a Riemannian manifold (M, g) is said to have totally skew-symmetric torsion, or skew torsion for short, if the (0, 3)-tensor field T defined by is a 3-form. The relation between ∇ and the Levi-Civita connection ∇ g is then given by It is well-known that any Sasaki manifold (M, ϕ, ξ, η, g) admits a characteristic connection, i. e. a unique metric connection ∇ with skew torsion such that ∇η = ∇ϕ = 0. Its torsion is given by T = η ∧ dη [FI02]. As a consequence, a 3-Sasaki manifold (M, ϕ i , ξ i , η i , g) cannot admit any metric connection with skew torsion such that ∇η i = ∇ϕ i = 0 for every i = 1, 2, 3. By relaxing the requirement on the parallelism of the structure tensor fields in a suitable way, one can define a large class of almost 3-contact metric manifolds, called canonical, including 3-(α, δ)-Sasaki manifolds, and thus 3-Sasaki manifolds. Any 3-(α, δ)-Sasaki manifold (M, ϕ i , ξ i , η i , g) is canonical, in the sense that it admits a unique metric connection ∇ with skew torsion such that for every even permutation (ijk) of (123), where β = 2(δ − 2α). The covariant derivatives of the other structure tensor fields are given by If δ = 2α, then β = 0 and the canonical connection parallelizes all the structure tensor fields. Any 3-(α, δ)-Sasaki manifold with δ = 2α is called parallel. Notice that this is a positive 3-(α, δ)-Sasaki manifold.
The torsion T of the canonical connection is given by where Φ H i = Φ i + η jk ∈ Λ 2 (H) is the horizontal part of the fundamental 2-form Φ i . Here we put η jk := η j ∧ η k and η 123 := η 1 ∧ η 2 ∧ η 3 . In particular, for every X, Y ∈ X(M ), (1.4) The symbol i,j,k S means the sum over all even permutations of (123). The torsion of the canonical connection satisfies ∇T = 0. The curvature properties of 3-(α, δ)-Sasaki manifolds will be discussed in detail in a separate publication [ADS21]. We cite from there without proof the following special result that will be needed in the following section. It is a side result of a lengthy and non-trivial, but otherwise straightforward computation.
). The curvature tensor R of the canonical connection of a 3-(α, δ)-Sasaki manifold satisfies for any X, Y, Z ∈ H and i, j, k, l = 1, 2, 3 the identities where in the last two identities (ijk) is an even permutation of (123).

The Riemannian submersion over a quaternionic Kähler base 2.1 The canonical submersion
In [CMS18] the authors discuss the geometry of Riemannian manifolds admitting metric connections ∇ τ with parallel skew torsion τ and reducible holonomy. This applies, in particular, to the canonical connection of 3-(α, δ)-Sasaki manifolds. We shortly recall their notation. Suppose the tangent space T M decomposes under the action of the holonomy group Hol of ∇ τ into a sum of irreducible representations v 1 , . . . , v r , h 1 , . . . , h s . Here an irreducible submodule is called vertical, adequately denoted by v j , if the subspace of hol acting purely on v j is trivial. Conversely, a subspace h a is called horizontal if the subspace k a = so(h a ) ∩ hol = {0} of hol acting purely on h a is non-trivial.
We need a slight generalization of the results obtained in [CMS18]. Suppose the tangent space decomposes into T M = v 1 ⊕· · ·⊕v r ⊕h 1 ⊕· · ·⊕h s as before. Let T M = V Γ ⊕H Γ be a decomposition such that for some subset Γ ⊂ Γ 0 = {1, . . . , r}. Suppose further that for this decomposition the projection of τ onto the space This condition turns out to be sufficient to prove Lemma 3.7-3.10 and Remark 3.11 from [CMS18]. We obtain b) there exists a 3-form σ ∈ Λ 3 N satisfying π * σ = pr Λ 3 HΓ τ , c) ∇ σ := ∇ gN + 1 2 σ defines a connection with parallel skew torsion σ on N . In particular, we have for the horizontal lifts X, Y ∈ T M of the vectors fields X, Y ∈ T N .

To a Riemannian submersion one assigns the O'Neill tensors
Here the subscripts denote projection on the respective subspaces. For the submersion above A and T simplify: The first expression follows directly. The identity T = 0 is then an immediate consequence of condition (2.2).
The vanishing of T does not come as a surprise since it is equivalent to the fibers being totally geodesic.
We now discuss the situation for 3-(α, δ)-Sasaki manifolds. By (1.2) the holonomy representation of the canonical connection ∇ of a 3-(α, δ)-Sasaki manifold splits into the horizontal and vertical subspaces H and V. In the non-parallel case V is irreducible, in the parallel case it decomposes into 3 trivial 1-dimensional representations. In either case the curvature properties stated in Proposition 1.2.1 allow us to prove: Lemma 2.1.1. The vertical distribution V of a 3-(α, δ)-Sasaki manifold is vertical with respect to the above notation.
Proof. By the Ambrose-Singer Theorem the holonomy algebra hol of the holonomy group Hol(p) at a point p is given by where P γ denotes parallel transport along γ and R(X, Y ) ∈ so(T q M ) the curvature operator. The horizontal and vertical distribution are invariant under parallel transport with respect to the canonical connection. Thus, we may assume γ to be trivial when investigating the holonomy action on these distributions. By (1.5) we know that the holonomy is only non-trivial if X, Y ∈ V or X, Y ∈ H. In the first case (1.6) and (1.7) show that every element of hol acting non-trivially on V must also act non-trivially on H. The action of an element R(X, Y ), X, Y ∈ H, on V is again given by (1.7). Any such element of hol acts non-trivially on V if β = 0 and Φ i (X, Y ) = 0 for some i = 1, 2, 3. In this case R(X, Y ) is also a non-trivial operator on H by (1.8).
Definition 2.1.1. We will call π : M → N the canonical submersion of a 3-(α, δ)-Sasaki manifold. We observe that the canonical submersion is, indeed, an almost contact metric 3-submersion in the sense of [Wa84], although we never make explicity use of this property (our formulas are much more detailed than the general results obtained therein).

The quaternionic Kähler structure on the base
We give a preliminary lemma needed to prove that the base of the canonical submersion admits a qK structure. Recall that a basic vector field on M is a horizontal vector field which is projectable, that is π-related to some vector field defined on N . If X ∈ T N , the horizontal lift of X is the unique basic vector field X ∈ T M such that π * X = X.
Lemma 2.2.1. For any vertical vector field X ∈ V and for any basic vector field Y ∈ H we have Proof. We first use the identity g(∇ g X Y, Z) = − 1 2 g([Y, Z], X) for any vector fields X ∈ V, Y, Z ∈ H, with Y and Z projectable, of a Riemannian submersion [Pe06,Proposition 13]. Note that the horizontal and vertical distributions of the Riemannian submersion agree with the same notion in the 3-(α, δ)-Sasaki setting. Further, we make use of Lemma 1.2.1 to obtain Theorem 2.2.1. The base N of the canonical submersion π : M → N of any 3-(α, δ)-Sasaki manifold M carries a quaternionic Kähler structure given by where s : U → M is any local smooth section of π. The covariant derivatives of the almost complex structuresφ i are given by Proof. Let s be a local section of the canonical submersion π : M → N , hence π * • s * = id and is an even or odd permutation of (123). This showsφ 2 i = −id andφ iφj = ±φ k . Finally, by means of (2.4) and (1.2), we show that the quaternionic structure is parallel. First By the properties of any Riemannian submersion we have that (π * (ϕ i (s * Y ))) = (ϕ i (s * Y )) H wherever the right side is defined, that is on the image s(N ) ⊂ M . Thus, we take the covariant derivatives in the direction of s * X resulting in a vertical correction termX = X − s * X ∈ V.
Recall that ∇ and ϕ i preserve the horizontal and the vertical distribution. Using Lemma 2.2.1, we obtain For the second summand, the horizontal projection is given by Recombining both identities we obtain Here we used the defining identity (1.2) of the canonical connection for any even permutation (ijk) of (123). Therefore, the quaternionic structure is parallel and N is quaternionic Kähler.
Remark 2.2.1. A priori the quaternionic structure may depend on the chosen section s. Indeed, the individual almost complex structuresφ i vary with s. However, following the work of P. Piccinni and I. Vaisman [PV01], one can see that the quaternionic structure is preserved under the Bott connectionD : This implies that the quaternionic structure is projectable and, thus, independent of choices.
3 Construction of non-degenerate homogeneous 3-(α, δ)-Sasaki manifolds For homogeneous 3-(α, δ)-Sasaki manifolds the canonical submersion is invariant. Hence, the base N is a homogeneous qK space. In the non-degenerate case Theorem 2.2.2 shows that N is a homogeneous quaternionic Kähler space of non-vanishing scalar curvature. There are two families of such spaces known: Compact qK symmetric spaces, named Wolf spaces, their non-compact duals and Alekseevsky spaces. The latter are homogeneous qK spaces admitting a solvable transitive group action. D. Alekseevsky conjectured that all homogeneous qK spaces with negative scalar curvature are Alekseevsky spaces [Ale75]. In particular, the class of non-compact qK symmetric spaces is included in the class of Alekseevsky. We will give independent constructions of homogeneous 3-(α, δ)-Sasaki manifolds over symmetric base spaces and such fibering over Alekseevsky spaces.

Homogeneous 3-(α, δ)-Sasaki manifolds over symmetric quaternionic Kähler spaces
Let G/G 0 be a real symmetric space, i.e. g = g 0 ⊕ g 1 with [g i , g j ] ⊂ g i+j on the level of Lie algebras. Suppose there exists a connected subgroup H ⊂ G 0 such that g 0 splits into a direct sum of Lie algebras g 0 = h ⊕ sp(1). Finally, assume that g C 1 = C 2 ⊗ C W , for some h C -module W of dim C W = 2n, and the adjoint action of g C 0 is given by where sp(1) C = su(2) C = sl(2, C) acts by multiplication on C 2 . We will call (G, G 0 , H) generalized 3-Sasaki data. b) Consider the homogeneous space M = G/H. The assumptions above imply that g = h ⊕ m with m = sp(1) ⊕ g 1 is a reductive decomposition. We rename the spaces V = sp(1) and H = g 1 to express their role as vertical and horizontal subspaces of a 3-(α, δ)-Sasaki manifold via T p M ∼ = m. For clarity we restate the bracket relations between all these spaces. We have g = h ⊕ V ⊕ H, where h and V are commuting subalgebras. Thus they form the joint subalgebra h ⊕ V = g 0 ⊂ g. The full set of commutator relations is In particular, both V and H are h-invariant.
c) Since G/G 0 is a symmetric space there exists a dual symmetric space G * /G 0 for every generalized 3-Sasaki data (G, G 0 , H). The Lie algebras can then be identified as It is then clear that (G * , G 0 , H) is generalized 3-Sasaki data as well. This yields pairs of compact and non-compact generalized 3-Sasaki data. For clarity we will denote the compact top Lie group by G and the non-compact one by G * . d) By [DOP18] any 3-Sasaki data gives rise to a homogeneous 3-Sasaki manifold. They were completely determined in [BGM94] by the fact that they are fiber-bundles over the quaternionic Kähler base space G/G 0 . The non-compact G * are thus given as the isometry group of the non-compact quaternionic Kähler symmetric spaces [Be87,p. 409]. Alltogether, we obtain     , g| H = −κ 8αδ(n + 2) , V ⊥ H. (1), where the σ i are the elements of sp(1) = su(2) given by Define endomorphisms ϕ i ∈ End h (m) for i = 1, 2, 3 by Together with η i = g(ξ i , ·) the collection (G/H, ϕ i , ξ i , η i , g) defines a homogeneous 3-(α, δ)-Sasaki structure.
Before we proceed with the proof, we collect some observations.
c) Usually the real representation g 1 of h will be irreducible and will only become reducible when complexified, thus we cannot describe the action of V = sp(1) on H easily, but from the complexified action we still find that the relations ad ξ 2 i = −δ 2 id and ad ξ i • ad ξ j = ±δad ξ k when (ijk) is an even, resp. odd permutation of (123) hold on H.
d) The Riemannian metric on H is a fixed multiple of the Killing form on g and thus the projection onto the symmetric orbit space G/H → G/G 0 is a Riemannian submersion. Indeed, this is the canonical submersion obtained in Theorem 2.2.1.

Sp(n)
by the action of Z 2 inside the fiber. Since the action is discrete these spaces cannot be discerned in the Lie algebra picture. Note that all relevant tensors are invariant under the Z 2 action and thus local results obtained for S 4n+3 = Sp(n+1) Sp(n) , resp. Sp(n,1) Sp(n) , remain true on RP 4n+3 and its non compact dual.

f) Since the metric is a multiple of the Killing form and the Killing form is ad-invariant [X, · ] will
be metric if it preserves H and V. This is precisely the case if X ∈ V. For X ∈ H, we compute Thus [X, · ] ∈ so(m) if and only if δ = 2α, i.e. we are in the parallel case. This is exactly the condition that our homogeneous space is naturally reductive. This can only occur if αδ > 0, i.e. we are in the positive case.
Proof (of Theorem 3.1.1). If G is compact, κ < 0. If G is of non-compact type, we have κ| V < 0 while κ| H > 0 by (3.1) . Thus, in both cases the given metric g is indeed positive definite.
On the contrary we have tr(ad ξ i • ad ξ j | H ) = tr(±ad ξ k | H ) = 0 as its trace on the complexification vanishes. And similar In any case tr(ad ξ i • ad ξ j ) = 0 and, hence, g(ξ i , ξ j ) = 0 if i = j.
Next we check that the endomorphisms ϕ i are metric almost complex structures on the complement to ξ i . Note that they vanish on their corresponding ξ i . Furthermore, Since H and V are invariant under ϕ i we check orthogonality on each component individually. On H use the associativity of κ to find and thus g(ϕ i X, ϕ i Y ) = −κ(ϕiX,ϕiY ) if (ijk), (ij ′ k ′ ) are according permutations of (123) and the left side vanishes whenever j or j ′ equals i. Next we check the compatibility conditions of the 3 almost contact metric structures. Suppose (ijk) is an even permutation of (123) then ϕ i ξ j = ξ k and together with the invariance of H under ϕ i we conclude η i • ϕ j = η k . Further, ϕ i ϕ j | H = 1 δ 2 ad ξ i • ad ξ j | H = 1 δ ad ξ k | H = ϕ k | H and on V we have We have thus shown that the given structure is a homogeneous almost 3-contact metric structure. It remains to show the 3-(α, δ)-Sasaki condition dη i = 2αΦ i + 2(α − δ)η j ∧ η k , for any even permutation (ijk) of (123). We show this case by case. Note that the last summand vanishes whenever either entry is in H. Let X ∈ H. Then, since ad ξ j X ∈ H, For X, Y ∈ H we use associativity of κ Finally, we have (3.2)

Negative homogeneous 3-(α, δ)-Sasaki manifolds over Alekseevsky spaces
In order to construct homogeneous 3-(α, δ)-Sasaki manifolds we recall the setup in the unified construction of Alekseevsky spaces due to V. Cortés [Co00]. Let q ∈ N. Set V = R 3,q the real vector space with signature (3, q). Let Cℓ 0 (V ) denote the even Clifford algebra over V . Depending on q mod 4 there exist exactly one or two inequivalent irreducible Cℓ 0 (V )-modules. Accordingly, let l ∈ N, if q ≡ 3 mod 4, or l + , l − ∈ N, if q ≡ 3 mod 4. Then set where W is the sum of l equivalent irreducible Cℓ 0 (V )-modules (or the sum of l + , l − irreducible Cℓ 0 (V )-modules if there are two inequivalent ones) and D a derivation with eigenvalue decomposition so(V ) ⊕ V ⊕ W and respective eigenvalues (0, 1, 1/2). The action of so(V ) on V is given by the standard representation and so(V ) acts on W via the isomorphism so(V ) ∼ = spin(V ) ⊂ Cℓ 0 (V ) e ∧ e ′ → − 1 2 ee ′ if e, e ′ are orthogonal. V commutes with itself and W . Finally the commutators [W, W ] are given by some non-degenerate so(V )-equivariant map Π : Λ 2 W → V where so(V ) acts on W as spin(V ). Remark 3.2.1. Note that Π is unique up to rescaling along the irreducible summands of W [Co00, Theorem 5]. This rescaling leads to an isomorphism of the Lie algebras g(Π) and g(Π ′ ) corresponding to two such maps Π and Π ′ . The isomorphism extends to an isomorphism of the 3-(α, δ)-Sasaki structures defined later on. Thus, we will ignore the ambiguity in Π from here on. Notation. On V = R 3,q fix an ONBê 1 ,ê 2 ,ê 3 , e 1 , . . . , e q with signature (+, +, +, −, . . . , −). Then with the identification so(V ) ∼ = Λ 2 V we also obtain a standard basis of the space so(V ) given by {ê i ∧ê j ,ê i ∧ e k , e k ∧ e l }i,j=1,2,3 k,l=1,...,3 . Denote σ i = 2ê k ∧ê j for any even permutation (ijk) of (123). Using the identification End(V ) = V ⊗ V * this implies [σ i ,ê j ] = 2ê k and [σ i , σ j ] = 2σ k where again (ijk) is an even permutation of (123).
We further set V = so(3) ⊂ so(3, q), H 0 the subspace generated by the elements D andê i + σ i and H 1 the subspace generated by e 1 , . . . , e q ∈ V and e i ∧ê j ∈ so(3, q).
The 4-dimensional spaces H 0 and e l , e l ∧ê j ⊂ H 1 will form the quaternionic subspaces inside so(V ) ⊕ V ⊕ RD ⊂ g. Accordingly, we show that they have the only commutators with non-trivial V-part.

Proof. The full list of commutators of basis vectors is
and finally where (ijk) is a permutation of (123) with ± indicating the sign of the permutation and l, m = 1, . . . , q with l = m. For the commutator By [Co00, Proposition 3] the adjoint action g r = RD ⊕ V ⊕ W ⊂ g is faithful. Thus, g is a subalgebra g ⊂ der(r). Set G the subgroup G ⊂ Aut(r) with Lie Algebra g. Let h = so(q) ⊂ so(V ) ⊂ g and H ⊂ G the corresponding connected subgroup. Then both G and H are closed subgroups of Aut(r). This follows from [Co00, Corollary 3] and the fact that H is closed in Spin 0 (V ) ⊂ G. In particular, G/H is a homogeneous space. We now define the desired negative 3-(α, δ)-Sasaki structure on M = G/H. Theorem 3.2.1. Let α, δ ∈ R with αδ < 0. Let G, H with Lie algebras g, h as above. Then m = V ⊕ H 0 ⊕ H 1 ⊕ W is a reductive complement to h in g. Set Define the almost complex structures ϕ i : m → m on V, H 0 , H 1 and W individually. For any permutation (ijk) of (123) with signature ± we set where ρ is the Clifford-multiplication on W . Define a scalar product g [e] by declaring the following vectors to be an orthonormal basis of V ⊕ H 0 ⊕ H 1 : On W we set the scalar product where , is the scalar product on V and (ijk) is any even permutation of (123). We set W orthogonal to V ⊕ H 0 ⊕ H 1 . Set η i = g(ξ i , · ) the dual to ξ i . Then (G/H, g, ξ i , η i , ϕ i ) defines a homogeneous 3-(α, δ)-Sasaki manifold.
Proof. We first note that the defined scalar product is positive definite and Spin(q)-invariant. This is clear on V ⊕ H 0 ⊕ H 1 and it is shown for b in [Co00, Theorem 1 and Proposition 9]. Thus, the scalar product extends to an invariant Riemannian metric on G/H. The invariance under H of the ξ i is obvious. For an invariant 3-a.c.m. structure, it remains to check that the ϕ i are invariant as well. Spin(q) acts trivial on V ⊕ H 0 and on H 1 by its adjoint action on e l ∈ R q ⊂ V . On W it acts by Clifford multiplication with vectors in R q twice, thus commuting with the Clifford multiplication defining the almost complex structures on W .
The endomorphisms ϕ i are compatible with the metric by definition on V ⊕ H 0 ⊕ H 1 and by Spin(q) · Spin(3)-invariance of b on W . Next we check the compatibility conditions of the 3 almost contact structures. Again on V ⊕ H 0 ⊕ H 1 this is a direct consequence of the definition and on W we have Finally we need to check the defining condition dη i = 2αΦ i + 2(α − δ)η j ∧ η k . By bilinearity it suffices to check it for any pair of two basis vectors individually. On V × V this is exactly the same computation as in the 3-(α, δ)-Sasaki structure over symmetric bases (compare (3.2)). Apart from V × V the equation reduces to dη i = 2αΦ i . Note that the left hand side reduces to checking the commutators. From Lemma 3.2.1 and the definition of the ϕ i we see that both sides vanish for all mixed terms regarding the decomposition V ⊕ H 0 ⊕ H 1 ⊕ W of the tangent space. Similarly on H 1 if the index l ofê i ∧ e l , respectively e l , is not the same both sides vanish. On H 0 × H 0 we compute In similar fashion for the remaining pairs in H 0 × H 0 and on H 1 × H 1 we have ê j ∧ e l ) = 2αΦ k (ê i ∧ e l ,ê j ∧ e l ) = 1 2δ for any even permutation (ijk) of (123). Finally, we look at W × W . Let w 1 , w 2 ∈ W and suppose Π(w 1 , w 2 ) = q r=1 a r e r + 3 s=1â sês . Then a r e r + 3 s=1â sês =â i δ .
This concludes the proof.

Examples
We begin with an example of the construction over a symmetric Wolf space.
Dimension parameters g h alternative description  Table 3.1, i.e. obtained by Theorem 3.1.1 over non-compact symmetric spaces. The list gets more intricate with higher dimension, in particular, there appear two inequivalent even Clifford modules for q = 3 beginning in dim 27 and for q ≥ 4 we have dim W q > 4. Further, observe that the symmetric base cases SU(2, 1)/U(1), G (2) 2 /SO(3) are not obtained by this construction. We now give more concrete descriptions of the Cℓ 0 (3, q)-modules W q for q = 0, 1, 2. Note that there are choices to be made though these lead to isomorphisms of the modules since all these modules are unique. Let R 3,q = e1, e2, e3, e 1 , . . . , e q , where eî have signature +1 while e i have signature −1. Then we have Cℓ 0 (3, 0) = H, Cℓ 0 (3, 1) = M 2 (C), Cℓ 0 (3, 2) = M 4 (R) realized as follows. Table 3.3 lists the cases q = 0 and q = 1, while Table 3.4 is devoted to the case q = 2. Table 3.3: Choice of Cℓ 0 (3, q)-representations for q = 0 and q = 1 The notation is as follow: We denote elements e ij = e i e j ∈ Cℓ 0 (V ) and analogous for the action of elements in Cℓ 0 (V ) of higher degree. The last line denotes the square of elements in the respective row, which are invariant of choices unlike the matrices itself. deg 0 : Table 3.4: Choice of Cℓ 0 (3, q)-representations for q = 2 With this we can find the map Π : Λ 2 W 2 → R 3,2 .
Next we describe the element 2D: (4.1) For the following theorem we need a similar statement to Lemma 1.2.1 for two fundamental vector fields. Note that even in the case when X, Y ∈ g are horizontal in the origin they fail to be horizontal in other points. Yet we have where Λ gN : H × H → H is the Nomizu map of the Levi-Civita connection on the homogeneous base of the canonical submersion.