Sectional nonassociativity of metrized algebras

The sectional nonassociativity of a metrized (not necessarily associative or unital) algebra is defined analogously to the sectional curvature of a pseudo-Riemannian metric, with the associator in place of the Levi-Civita covariant derivative. For commutative real algebras nonnegative sectional nonassociativity is usually called the Norton inequality, while a sharp upper bound on the sectional nonassociativity of the Jordan algebra of Hermitian matrices over a real Hurwitz algebra is closely related to the B\"ottcher-Wenzel-Chern-do Carmo-Kobayashi inequality. These and other basic examples are explained, and there are described some consequences of bounds on sectional nonassociativity for commutative algebras. A technical point of interest is that the results work over the octonions as well as the associative Hurwitz algebras.


Introduction
For an algebra pA, ˝q equipped with a metric, meaning a nondegenerate symmetric bilinear form, that is invariant, meaning hpx ˝y, zq " hpx, y ˝zq for all x, y, z P A, the sectional nonassociativity Kpx, yq of the h-nondegenerate subspace Spantx, yu Ă A is defined in [18] to be Kpx, yq " hpx ˝x, y ˝yq ´hpx ˝y, y ˝xq hpx, xqhpy, yq ´hpx, yq 2 .(1.1) (Because ˝need not be commutative or anticommutative, the order of x and y in (1.1) matters.)The sectional nonassociativity is analogous to the sectional curvature of the Levi-Civita connection of a pseudo-Riemannian metric, with the multiplication and its associator in place of the covariant derivative and its curvature.Section 3 describes in detail this analogy between the multiplication of an algebra an torsion-free connection on a tangent bundle.The notion itself, and its basic properties, are given in Section 4.
Quantitative bounds on the sectional nonassociativity of a metrized real algebra encompass apparently unrelated inequalities such as the Böttcher-Wenzel-Chern-do Carmo-Kobayashi inequality bounding the norm of a matrix commutator and the Norton inequality asserting the nonnegativity of the quadratic expression hpx ˝x, y ˝yq ´hpx ˝y, x ˝yq for certain metrized commutative algebras arising in the theory of vertex operator algebras whose most famous representative is the 196844-dimensional Griess algebra on which the monster finite simple group acts by automorphisms.
It is apparent from (1.1) that, for a commutative algebra, the Norton inequality may be rephrased as nonnegative sectional nonassociativity.On the other hand, if pA, r ¨, ¨s, hq is a metrized anticommutative algebra (for example, a Lie or Malcev algebra with h a multiple of its Killing form), (1.1)If h is positive definite, the right-hand side of (1.2) is bounded from above by hprx,ys,rx,ysq hpx,xqhpy,yq and, by the Cauchy-Schwarz inequality, this quantity is bounded from above.The Böttcher-Wenzel inequality asserts that for the algebra Mpn, Cq of n ˆn complex matrices metrized by the Frobenius norm this quantity is bounded from above by 2 (which is better than the constant obtained by applying Cauchy-Schwarz), and characterizes the case of equality.The Böttcher-Wenzel inequality can be interpreted as an upper bound on the sectional nonassociativity of the Lie algebra of n ˆn matrices Mpn, Cq with the usual matrix commutator.
For an anticommutative algebra, (1.2) is obviously nonnegative, with equality if and only if rx, ys " 0, so that characterization of the equality case in the lower bound amounts to characterization of commuting pairs of elements.On the other hand, for a commutative algebra, (1.1) can take both signs.These observations suggest looking for similar bounds for other metrized algebras, for example commutative algebras or other sorts of Lie algebras.
The extreme case of equal upper and lower bounds is that of constant sectional nonassociativity.This is closely related to the notion of projective associativity of commutative algebras studied in [18].Although it is not solved here, the classification of metrized algebras with constant sectional nonassociativity appears an accessible problem.However, if anisotropy of the metric is not assumed, examples abound and it is still not clear what form the classification will take even in the simplest case of vanishing sectional nonassociativity.Essentially by definition, associative and alternative algebras have vanishing sectional nonassociativity.In Section 5, it is shown that both a quadratic 2-step nilpotent Lie algebra and a metrized antiflexible algebra have vanishing sectional nonassociativity, and nontrivial examples of such algebras are given.In particular, Example 5.2 exhibits a Euclidean metrized antiflexible algebra with vanishing sectional nonassociativity whose underlying Lie algebra is sop3q (which has constant sectional nonassociativity 1 when metrized by the negative of its Killing form) and whose underlying commutative algebra has constant negative sectional nonassociativity.(It gives a similar example with underlying Lie algebra slp2, Rq.) Section 5 describes other examples of algebras having constant sectional nonassociativity.Example 5.3 explains that cross product algebras could be defined as antisymmetric metrized algebras having constant negative sectional nonassociativity.Example 5.6 describes a one-parameter family of pairwise nonisomorphic 3-dimensional metrized commutative algebras having constant sectional nonassociativities in p´8, 1{4s.
Section 6 gives examples of algebras for which there can be established bounds on the sectional nonassociativities.Example 6.1 recalls a result of Miyamoto shows that the Griess algebras of OZ vertex operator algebras satisfy the Norton inequality.In this context it would be interesting to establish upper bounds.Example 6.2 shows that symmetric composition algebras such as para-Hurwitz and Okubo algebras have sectional nonassociativity bounded between ´1 and 1.
The remainder of Section 6 is devoted to the proof of Theorem 6.3, which gives sharp upper and lower bounds on the sectional nonassociativities of a simple Euclidean Jordan algebra of rank at least 3 together with characterization of the cases of equality in the bounds.These algebras are simply the Jordan algebras Hermpn, hq of Hermitian matrices over the real Hurwitz algebras h P tR, C, H, Ou, where n ě 3, and, in the case of the octonions, O, attention is restricted to n " 3. Theorem 6.3 states that the sectional nonassociativity, Kpx, yq, defined in Definition 4.1, of the subspace spanned by linearly independent x, y P Hermpn, hq satisfies 0 ď Kpx, yq " K h,‹ px, yq ď n 2 , (1.3) and characterizes the subspaces for equality is obtained in the upper and lower bound.The lower bound is the Norton inequality.The sharp upper bound is deduced as a consequence of the Chern-do Carmo-Kobayashi inequality for the commutators of Hermitian matrices.This requires some argument to extend this inequality to the octonions that is formulated as Lemma 7.2; this extension may be of independent interest.These results are of interest from the point of view of applications of the Böttcher-Wenzel-Chern-do Carmo-Kobayashi inequality.This inequality is recalled in section 7 and it is shown that the Chern-do Carmo-Kobayashi inequality holds for Hermpn, Hq and partially for Hermp3, Oq.These refinements, Theorem 7.1 and the variant Lemma 7.2, valid over a real Hurwitz algebra, are used in the proof of Theorem 6.3 to estimate the sectional nonassociativity of the deunitalization of a simple Euclidean Jordan algebra.In a sense, this amounts to reinterpreting the Böttcher-Wenzel-Chern-do Carmo-Kobayashi inequalities as upper bounds on sectional nonassociativity.
The notion of sectional nonassociativity facilitates quantifications of nonassociativity both more refined and less exact than nonnegativity or nonpositivity, and these may in turn suggest refinements of the Böttcher-Wenzel inequality.More generally, the sectional nonassociativity is manifestly invariant under isometric algebra automorphisms and its relation with other similar quantities deserves to be explored more.See Remark 7.5 for further comments in this direction.
Not many examples of commutative algebras satisfying the Norton inequality are known and the description of such algebras is interesting and their classification, possibly with additional hypotheses, may be a possibly tractable problem.The main known examples are the Griess algebras of certain vertex operator algebras [43] and their subalgebras, as is explained next in Example 6.1, and, by Theorem 6.3, the simple Euclidean Jordan algebras, as is shown in Theorem 6.3.(It is known that most of the simple Euclidean Jordan algebras can also be obtained as Griess algebras of certain vertex operator algebras [3,35,68].)Further examples are the Majorana algebras introduced by A. A. Ivanov [28,29] which are defined by a list of axioms including the Norton inequality.See [65] for a family of examples depending on a parameter and further references.
More generally, the notion of sectional nonassociativity should help to organize the study of metrized algebras in the same manner that restriction on curvature quantities help to organize the study of Riemannian manifolds.Typically classes of algebras are defined by identities.From the point of view advocated here this is something like studying symmetric spaces (whose curvature tensors satisfy algebraic equalities).
Section 8 presents some general consequences for commutative algebras of bounds on sectional nonassociativity.Combined with the notion of sectional nonassociativity, results about the eigenvectors and eigenvalues of the structure tensor can be used to relate the existence and description of idempotents and square-zero elements to conditions involving sectional nonassociativity.Since an automorphism of the algebra must permute its idempotents and square-zero elements, such results yield results about the size of the automorphism group.For example, Theorem 8.1 shows that a Euclidean metrized commutative algebra has finite automorphism group if its sectional nonassociativity is negative or if its sectional nonassociativity is nonpositive and the algebra contains no trivial subalgebra.In the other direction, Lemma 8.5 shows that nonnegative sectional nonassociativity precludes existence of 2-nilpotents.The flavor of such results conforms broadly with the geometric analogy.For example, the automorphism group of a negatively curved Riemann surface is finite, while the (positively curved) sphere has a large continuous group of automorphisms.
The paper is structured as follows.Section 2 presents basic definitions related to metrized algebras.Section 3 shows that the analogy between multiplications and covariant derivatives has substance and presents results that help motivate the definition of sectional nonassociativity given in Section 4. Section 4 also describes the most basic properties of the sectional nonassociativity.Section 5 presents examples having constant sectional nonassociativity; these include Hurwitz algebras, cross-product algebras, antiflexible algebras, and some novel three-dimensional algebras.Section 6 describes examples satisfying bounds on sectional nonassociativity generalizing the Norton inequality; these include symmetric composition algebras, Griess algebras of OZ vertez operator algebras, and the simple Euclidean Jordan algebras.The proof of the upper bound on the sectional nonassociativity of simple Euclidean Jordan algebras given in Theorem 6.3 requires the Böttcher-Wenzel-Chen-do Carmo-Kobayashi inequality for matrices over a Hurwitz algebra.Section 7 describes this and addresses what happens over the octonions.Finally, Section 8 presents some general consequences for commutative algebras of bounds on sectional nonassociativity.

Preliminaries
Let k be a field.Everywhere in the paper there is assumed char k ffl 6, although sometimes assumptions on char k are stated explicitly for clarity.
An algebra means a finite-dimensional vector space A over k equipped with a bilinear multiplication ˝: A ˆA Ñ A. It need not be associative, commutative, or unital.The left and right multiplication endomorphisms L ˝, R ˝: A Ñ EndpAq are defined by L ˝pxqy " x ˝y " R ˝pyqx.
An algebra is metrized if it is equipped with a metric, meaning a nondegenerate symmetric bilinear form, h, that is invariant (also called associative), meaning that hpx ˝y, zq " hpx, y ˝zq for all x, y, z P A.
A Euclidean metrized algebra is a real algebra metrized by a positive definite metric.(Essentially all results in the paper stated for Euclidean metrized algebras remain true with a real-closed base field in place of R, but it would be distracting to state and prove them in this generality.)That an algebra be Euclidean metrized means that the adjoint of the left multiplication endomorphism L ˝pxq equals the right multiplication endomorphism R ˝pxq.Consequently, for a Euclidean metrized commutative algebra the multiplication endomorphisms L ˝pxq " R ˝pxq are self-adjoint with respect to a positive definite symmetric bilinear form.Proof.If u and v span A, then hpu ˝v, uq " hpu, u ˝vq " hpu ˝u, vq " hpv, u ˝uq " hpv ˝u, uq and, similarly, hpu ˝v, vq " hpv ˝u, vq.By the nondegeneracy of h this implies u ˝v " v ˝u, which suffices to show the claim.
The associator of an algebra is the trilinear map b 3 A ˚Ñ A defined by rx, y, zs " A ˝px, yqz " pL ˝px ˝yq ´L˝p xqL ˝pyqqz " px ˝yq ˝z ´x ˝py ˝zq. (2.2) The map A ˝px, yq P EndpAq is the (left) associator endomorphism.

Example 2.2.
A commutative algebra is Jordan if rx ˝x, y, xs " 0 for all x, y, P A. For a Jordan algebra, the bilinear form tr L ˝px ˝yq is invariant, so it is metrized if this bilinear form is nondegenerate.A real Jordan algebra is Euclidean if it is formally real, meaning that a sum of squares equals zero if and only if each of the elements squares is zero.A real Jordan algebra is Euclidean as a Jordan algebra if and only if tr L ˝px ˝yq is positive definite [30].Ÿ Example 2.3.A composition algebra is an algebra pA, ˝, qq with a nondegenerate quadratic form q that is multiplicative meaning qpx ˝yq " qpxqqpyq for all x, y P A. A Hurwitz algebra is a unital composition algebra pA, ˝, e, qq.By the Hurwitz theorem, a real Hurwitz algebra is one of the real field R, the complex field C, the quaternions H, or the octonions O.For background on composition and Hurwitz algebras see [14,32,59].A Hurwitz algebra carries the nondegenerate symmetric bilinear form hpx, yq " qpx `yq ´qpxq ´qpyq obtained from q by linearization.By the nondegeneracy and multiplicativity of q, hpe, eq " 2qpeq " 2. The standard involution of pA, ˝, e, qq is the algebra antiautomorphism defined by x " hpx, eqe ´x.Its fixed point set is the subalgebra generated by e and is isomorphic to R. Taking z " e yields hpx, yqe " xȳ `y x " hpx, ȳqe.A Hurwitz algebra equipped with its standard involution is a metrized algebra with involution in the sense of [2], meaning that h is involutively invariant in the sense that hpx, z ˝ȳq " hpx ˝y, zq " hpy, x ˝zq for all x, y, z P A. Note that, except when h " k, h is not invariant, so ph, ˝, hq is not metrized as an algebra.Ÿ

The analogy between multiplications and connections
The following analogy between algebras and connections illuminates the point of view taken here.The multiplication ˝is analogous to a torsion-free connection on a tangent bundle, and the associator corresponds to (the negative of) the second covariant derivative.In fact, a torsion-free connection on a tangent bundle can be regarded as a Lie-admissible product on the space of vector fields, and in that setting the sectional nonassociativity defined in Section 4 is simply the sectional curvature.However that setting differs from that considered here in several ways.First, the space of vector fields is infinite-dimensional.Second, the covariant derivative is a C 8 -module map in its first argument.A general context encompassing both finite-dimensional algebras and connections on tangent bundles is algebroids, but this is not discussed here.
The notions suggested by the multiplication-connection analogy prove surprisingly robust and provide principles for organizing the zoo of general algebras different from those that arise from classical points of view.In particular, they offer possibilities for quantifying nonassociativity that turn out to be quite useful.
This idea has been discussed in similar terms in [26] in the context of Lie-admissible algebras and has roots in foundational articles on left-symmetric algebras [60,63].In [8] there are considered algebras satisfying identities like those treated in this section from the more sophisticated point of view of Koszul duality of operads.The unrelated duality given by the adjoint multiplication considered here is motivated by considerations from affine differential geometry [19].These ideas are expounded in the more limited context of metrized commutative algebras in [18,20].
The analogy is implemented in practice in the following way.The space of connections on a bundle is an affine space.A reference connection is fixed and then any other connection is identified with a tensor.Any expression involving this connection involves some derivatives and some purely algebraic part expressible in terms of this tensor.The purely algebraic expression make sense also for the structure tensor of an algebra.For example, the covariant derivative of a metric g with respect to a torsion-free affine connection ∇ is p∇ X gqpY, Zq " XpgpY, Zqq ´gp∇ X Y, Zq ´gpY, ∇ X Zq.The corresponding expression for a metric g on an algebra pA, ˝q is ´gpx ˝y, zq ´gpy, x ˝zq.The vanishing of this expression is analogous to the metric tensor being parallel.This kind of invariance should probably be properly formulated as a cohomological condition, but it is not clear how to give sense to this last claim when working in the generality considered here.
In the analogy, the underlying bracket is regarded as background information of a topological rather than geometric nature, analogous to the Lie algebra of vector fields on a smooth manifold.
Applying the connection-algebra analogy to the curvature of an affine connection yields the following definitions.
Decomposing the product ˝by symmetries leads to two underlying algebras, from which pA, ˝q can be reconstructed (except when char k " 2).The underlying bracket (or commutator) of the algebra pA, ˝q is the commutator defined by rx, ys " x ˝y ´y ˝x and pA, r ¨, ¨sq is the underlying antisymmetric algebra.The underlying commutative algebra of an algebra pA, ˝q is A equipped with the product x d y " x ˝y `y ˝x.
The multiplication ˝t adjoint to ˝is that defined by x ˝t y " ´y ˝x.It is the negative of the opposite multiplication.By definition, that L ˝t " ´R˝a nd R ˝t " ´L˝.The sign in the definition of the adjoint multiplication ˝t is chosen so that ˝and the adjoint product ˝t have the same underlying bracket.
The The adjoint curvature of ˝is the curvature of the adjoint multiplication ˝t and vice-versa.For any tensorial object constructed from a multiplication ˝there is a corresponding tensorial object constructed in the same way from the adjoint multiplication ˝t, and it is denoted by putting a ¯over the notation indicating the object associated with ˝.This convention is consistent with the notation r and r.
The algebra has self-adjoint curvature if r " r and anti-self-adjoint curvature if r " ´r.Any left-symmetric algebra that is not right-symmetric gives an example of an algebra pA, ˝q not isomorphic to its adjoint pA, ˝tq.More generally, an algebra that does not have self-adjoint curvature is not isomorphic to its adjoint algebra.Ÿ Example 3.4.An awkward aspect of the conventions here is that the underlying bracket of an anticommutative algebra is twice the original bracket, and this needs to be remembered when interpreting (3.1).
As a Lie algebra pg, r ¨, ¨sq is flexible, it has self-adjoint curvature, and from (3.1) it follows that rpx, yq " rpx, yq " ´ad g prx, ysq " ´rad g pxq, ad g pyqs.Consequently any 2-step nilpotent Lie algebra provides an example of a nonassociative algebra for which both r and r vanish identically. Ÿ for all x, y, z P A (the validity of the last equality needs char k ‰ 2).
Lemma 3.1.If char k ‰ 2, the following conditions on a k-algebra pA, ˝q are equivalent.
(3) pA, ˝q is flexible.Over a field of any characteristic, a flexible algebra satisfies (1) and (2) and has self-adjoint curvature.
Proof.By definition, pA, ˝q is flexible if and only if rR ˝pxq, L ˝pxqs " 0 for all x P A. By (3.3) this implies (1) and is implied by (1) if char k ‰ 2. Polarization shows any flexible algebra satisfies (2), and if char k ‰ 2, then the validity of (2) implies pA, ˝q is flexible (taking y " x in (2)).By (2) and (3.4), a flexible algebra has self-adjoint curvature.Lemma 3.2.If char k ‰ 2, a k-algebra pA, ˝q is flexible if and only if it has self-adjoint curvature and A ˝pxq is a derivation of the underlying commutative algebra pA, dq for all x P A.
Proof.This is immediate from Lemma 3.1 and (3.2).
Example 3.5.By [2, Proposition 1], the right-hand side of (3.4) vanishes for any three elements in the subspace of a structurable algebra fixed by its involution.Ÿ The analogue of the algebraic Bianchi identity for a curvature tensor is the vanishing of the complete antisymmetrization of rpx, yqz.This need not occur in general for, as is the case for an affine connection, there is an obstruction related to compatibility with an underlying Lie bracket.
An algebra pA, ˝q is Lie-admissible if the underlying bracket r ¨, ¨s satisfies the Jacobi identity, so that pA, r ¨, ¨sq is a Lie algebra.That pA, ˝q be Lie-admissible is equivalent to the vanishing of the completely antisymmetric tensor (The notation Cycle 1,2,3 indicates the sum over the cyclic permutations of the indicated indices.)The validity of the identity ´Cycle 1,2,3 rpx 1 , x 2 qx 3 " Cycle prx 1 , x 2 , x 3 s ´rx 2 , x 1 , x 3 sq " Cyclerrx 1 , x 2 s, x 3 s in any algebra is noted in [48,Equation 2.19].That the vanishing of the first expression on the second line of (3.5) is equivalent to Lie admissibility is stated in [40,Example 6].
Note that t " t.The final two equalities of (3.5) show that the identity t " 0 is analogous to the algebraic Bianchi identity for a torsion-free connection, so the tensor t should be regarded as the algebraic analogue of the torsion of an affine connection.
A straightforward computation shows t is the self-adjoint part of rA ˝pxq, L ˝pyqs ´L˝p A ˝pxqyq (3.6) which vanishes for all x, y P A if and only if A ˝pxq is a derivation of pA, ˝q for all x P A.
Example 3.6.It is immediate from (3.5) that t " 0 for an antiflexible algebra, so an antiflexible algebra is Lie-admissible.Ÿ The analogue of the covariant derivative of the curvature tensor, p∇ X RqpY, Zq, is rL ˝pxq, rpy, zqs ŕpL ˝pxqy, zq ´rpy, L ˝pxqzq.The identities (3.7) are analogous to the differential Bianchi identity, and reinforce the idea that t should be viewed as something like the torsion of ˝.
Proof.A computation using (3.1), the Jacobi identity in EndpAq, the identity rx ˝y, zs ´rx, z ˝ys " rx, y, zs ŕz, y, xs, and (3.5) shows the first identity of (3.7).Taking ˝t in place of ˝and recalling t " t yields the second identity of (3.7).
An algebra pA, ˝q is associator-cyclic if it satisfies the identity 0 " rx, y, zs `rz, x, ys `ry, z, xs " pA ˝px ˝yq ´A˝p xqL ˝pyq ´A˝p yqR ˝pxqq z, (3.8) for all x, y, z P A. (There seems to be no standard terminology for algebras satisfying (3.8), and associatorcyclic seems reasonably evocative terminology.)[45]).For a k-algebra pA, ˝q the following are equivalent.
(2) For all x P A, A ˝pxq " L ˝pxq ´R˝p xq is a derivation of pA, ˝q.
Proof.For any algebra there holds Cycle x,y,z rx, y, zs " prA ˝pxq, L ˝pyqs ´L˝p A ˝pxqyqq z `rL ˝pxq, R ˝pzqsy.(3.9) It is immediate from (3.9) that if pA, ˝q is flexible and associator-cyclic, then it satisfies (2).If pA, ˝q satisfies (2), then ´rR ˝pxq, L ˝pxqs " rA ˝pxq, L ˝pxqs " L ˝pA ˝pxqxq " 0 for all x P A, so pA, ˝q is flexible, and with (3.9) it follows that pA, ˝q is associator-cyclic.This proves the equivalence of ( 1) and ( 2).The equivalence of ( 2) and ( 3) is immediate from (3.6) and its adjoint relation.
Remark 3.8.The essential content of Lemma 3.5 is due to Okubo and Myung [44,45,50].It is formulated here in a manner consistent with the perspective taken here.The equivalence of ( 1) and ( 2) and that these imply ( 3) is also proved, with different terminology, in [27] (see also [54,Propositions 1 and 2]).(In [54] an algebra is called admissible Poisson if there holds (2) and weakly associative if there vanishes the right-hand side of (3.9) (so it satisfies (3) of Lemma 3.5).)Ÿ Example 3.9.By the Jacobi identity, the associator of a Lie algebra pg, r ¨, ¨sq is rx, y, zs " ry, rx, zss and a Lie algebra is associator-cyclic.This shows an associator-cyclic, flexible algebra need not be commutative.Ÿ Example 3.10.By [57] an algebra obtained from a flexible algebra by the Cayley-Dickson process is flexible.An example of a flexible algebra that is not Lie-admissible (so not associator-cyclic) is the imaginary octonions im O equipped with the commutator bracket rx, ys " xy ´yx.(Although im O is Malcev-admissible [45].)Ÿ

Sectional nonassociativity
This section introduces quantitative notions of nonassociativity for metrized algebras.The sectional nonassociativity was introduced in [18] for metrized commutative algebras.
In the first part of this section the base field k has characteristic not equal to 2 or 3. Let A be an n-dimensional k-vector space with a metric h.Define apx, y, z, wq " ´apy, x, z, wq " ´apx, y, w, zq " apz, w, x, yq for all x, y, z, w P A + , MCpAq " ta P S 2 Λ 2 A ˚: apx, y, z, wq `apy, z, x, wq `apz, x, y, wq " 0 for all x, y, z, w P Au. (4.1) The subspace MCpAq comprises tensors of metric curvature tensor type.An element a P S 2 Λ 2 A ˚can be viewed as the bilinear form on Λ 2 A defined on decomposable elements by apx ^y, z ^wq " 4apx, y, z, wq, so determines an associated quadratic form on Λ 2 A. Endow b 4 A ˚and its subspaces with the metrics given by complete contraction with the metric h.If char k ‰ 3, S 2 Λ 2 A ˚is isomorphic to the orthogonal direct sum MCpAq ' Λ 4 A ˚, and the orthogonal projections onto the factors, P : Ppaqpx, y, z, wq " 1 3 p2apx, y, z, wq ´apy, z, x, wq ´apz, x, y, wqq , Qpaqpx, y, z, wq " 1 3 papx, y, z, wq `apy, z, x, wq `apz, x, y, wqq . (4. 2) The tensors a and Ppaq do not necessarily determine the same quadratic form on Λ 2 A, but the quadratic forms they determine agree on decomposable elements of Λ 2 A.
Lemma 4.1.Suppose char k ffl 6.Let h be a metric on the k-vector space A and let P : S 2 Λ 2 A ˚Ñ MCpAq be the orthogonal projection with respect to the inner products given by complete contraction with h. in which x ¨, ¨y denotes the pairing between dual vector spaces.Because ω ^ω " 0 if ω is decomposable, this shows that the restrictions to decomposable 2-forms of the quadratic forms determined by a and Ppaq coincide, and, moreover, that if Qpaq " 0, then these quadratic forms are equal.
On the other hand, that the quadratic forms determined on Λ 2 A by a, b P S 2 Λ 2 A ˚coincide on decomposable elements means that c " a ´b P S 2 Λ 2 A ˚satisfies 4cpx, y, x, yq " xc, px ^yq b px ^yqy " 0 for all x, y P A. Linearizing in x and y and using the symmetries of c yields 0 " cpx, y, z, wq `cpz, y, x, wq `cpx, w, z, yq `cpz, w, x, yq " 2cpx, y, z, wq ´2cpy, z, x, wq (4.4) for all x, y, z, w P A. Hence cpx, y, z, wq " cpy, z, x, wq " cpz, x, y, wq, and in (4.2) this yields Ppaq ´Ppbq " Ppcq " 0.
Let pA, ˝, hq be a metrized algebra of dimension at least 2. If x, y P A span a two-dimensional subspace V Ă A and x " ax `by and ȳ " cx `dy span the same subspace V, then hpx, xqhpȳ, ȳq ´hpx, ȳq 2 " pad ´bcq 2 `hpx, xqhpy, yq ´hpx, yq 2 ˘, (4.10) hpx ˝x, ȳ ˝ȳq ´hpx ˝ȳ, x ˝ȳq " pad ´bcq 2 phpx ˝x, y ˝yq ´hpx ˝y, y ˝xqq , (4.11) from which it follows that (4.11) divided by (4.10) is well-defined and depends only the subspace V, provided the denominator is non-zero, which is the case as long V is h-nondegenerate.(A symmetric bilinear form h on a two-dimensional k-vector space V is nondegenerate if and only if hpx, xqhpy, yq ´hpx, yq 2 ‰ 0 for any basis tx, yu of V; this is a special case of the statement that a symmetric bilinear form is nondegenerate if and only if the determinant of its Gram matrix with respect to any basis is nonvanishing [ The proviso that Spantx, yu be h-nondegenerate is automatic when h is anisotropic.By definition, the sectional nonassociativity is invariant under isometric algebra isomorphisms of metrized algebras.
The dependence of K " K ˝,h on ˝and h is indicated with subscripts when helpful.For r, s P 9 R, K s˝,rh " s 2 r ´1K ˝,h , where s˝means the multiplication sx ˝y.
Given an algebra pA, ˝q, with underlying anticommutative and commutative products r ¨, ¨s and d, a straightforward computation shows that that h is invariant with respect to both of r ¨, ¨s and d, consequently the sectional nonassociativities K r ¨, ¨s,h px, yq and K Proof.That the coincidence of ´1 2 pr 5 `r 5 q and cph hq on decomposable elements of Λ 2 A implies constant sectional nonassociativity is immediate from the definition (4.13).
The Grassmannian Grp2, Aq of two-dimensional subspaces of A is a projective variety via the Plücker embedding Grp2, Aq Ñ PpΛ 2 Aq sending ktx, yu P Grp2, Aq to the span ktx ^yu.The homogeneous functions H, K : Λ 2 A Ñ k defined by Hpx ^yq " hpx, xqhpy, yq ´hpx, yq 2 and Kpx ^yq " ´1 2 pr 5 `r 5 qpx, y, x, yq " hpx ˝x, y ˝yq ´hpx ˝y, y ˝xq are quadratic polynomials that restrict to the image of the Plücker embedding of the Grassmannian Grp2, Aq of two-dimensional subspaces of A. If k is infinite, then the intersection of any two nonempty Zariski open subsets of Grp2, Aq is nonempty.That the sectional nonassociativity be constant equal to c is the same as assuming cH ´K vanishes on Grp2, AqztH " 0u.As the complement Grp2, AqztH " 0u is infinite, if cH ´K vanishes on Grp2, AqztH " 0u, it vanishes on Grp2, Aq.Lemma 4.4.Let pA, ˝, hq be a metrized algebra over an infinite field of characteristic not equal to 2 or 3.
(1) pA, ˝, hq has constant sectional nonassociativity c if and only if Ppr 5 `r 5 q " 2ch h.
( If there holds (1), then the last term of (4.18) equals cph hqpx, y, x, yq, so (4.18) shows Kpx, yq " c.On the other hand, by Lemma 4.3, that Kpx, yq " c for all linearly independent x, y P A ˚means that the quadratic forms determined on Λ 2 A by ´1 2 pr 5 `r 5 q and cph hq coincide on decomposable elements of Λ 2 A. By Lemma 4.1 this implies (1).If ˝is Lie-admissible, then, by Lemma 4.2, Ppr 5 `r 5 q " r 5 `r 5 , so in this case (1) is equivalent to (2).If ˝is moreover flexible, then r " r, so ( 2) is equivalent to (3).
Example 4.1.Evidently the sectional nonassociativity of a metrized associative algebra is identically zero.More generally, the same is true for an alternative algebra.An algebra pA, ˝q is alternative if rx, x, ys " 0 and rx, y, ys " 0 for all x, y P A. By (4.13), the sectional nonassociativity of a metrized alternative algebra is identically zero.This can also be seen as a consequence of the theorem of E. Artin (see [58, p. 18]) that shows that the subalgebra generated by any two elements of an alternative algebra is associative.
For example, any Hurwitz algebra is alternative, so has vanishing sectional nonassociativity.This observation illustrates the limitations of the analogy between the associator and the curvature tensor, for the octonions are alternative but not associative.Ÿ In the rest of this section it is supposed the base field is R. A Euclidean metrized algebra pA, ˝, hq has positive, nonnegative, zero, etc. sectional nonassociativity if the given qualifier is valid for every two-dimensional subspace of A.
For a Euclidean metrized algebra, by the Cauchy-Schwarz inequality, the numbers UpA, ˝, hq " sup x,yPA:x^y‰0 are finite and invariant under isometric automorphisms of pA, ˝, hq.Their estimation for any well delineated class of metrized algebras is a basic problem.
Such an estimate is most interesting when the invariant metric h is determined in some canonical manner by the algebra structure, as is the case for the Killing form τ ˝px, yq " tr L ˝pxqL ˝pyq of a Lie or Malcev algebra or the form tr L ˝px ˝yq on a Jordan algebra.
Lemma 4.5 shows that taking direct sums of Euclidean metrized commutative algebras preserves conditions such as nonpositive and nonnegative sectional nonassociativity.
For readability, in the proofs of Lemmas 4.5 and 4.
with equality if and only if x " x 1 `x2 and y " y 1 `y2 are linearly dependent.Let x i P A i be the h-orthogonal projection of x P A. By (4.21), r i " |xi^yi| 2 |x^y| 2 is contained in r0, 1s, and, by the orthogonality of the ideals A 1 and A 2 , it follows that for linearly independent x, y P A, Applying to (4.22) the inequalities minta, b, cu ď ar 1 `br 2 `cp1 ´r1 ´r2 q ď maxta, b, cu, valid for any r 1 , r 2 P r0, 1s, with pa, b, cq " pm 1 , m 2 , 0q or pa, b, cq " pM 1 , M 2 , 0q yields (4.20).
It is more difficult to relate the sectional nonassociativity of a tensor product to the sectional nonassociativities of its factors, but in some cases something can be said.
The tensor product of metrized algebras pA i , ˝i, h i q, i P t1, 2u, is the vector space A 1 b A 2 with the product defined by pa 1 b b 1 q ˝pa 2 b b 2 q " pa 1 ˝1 b 1 q b pa 1 ˝2 b 2 q and extending bilinearly, and the metric defined by hpa 1 b b 1 , a 2 b b 2 q " h 1 pa 1 , b 1 qh 2 pa 1 , b 2 q and extending bilinearly.Lemma 4.6.For i P t1, 2u let pA i , ˝i, h i q be Euclidean metrized commutative algebras.For a i , āi P A i , if K A1,h1 pa 1 , ā1 q and K A2,h2 pa 2 , ā2 q are both nonnegative, then K A1bA2,h1bh2 pa 1 b a 2 , ā1 b ā2 q is nonnegative.
from which the claim follows.

Examples of algebras with constant sectional nonassociativity
This section presents examples of metrized algebras having constant sectional nonassociativity.
Example 5.1.By Example 3.4 a 2-step nilpotent Lie algebra satisfies r " r " 0. If such a Lie algebra is equipped with an invariant metric, then its sectional nonassociativity vanishes identically.In general a 2-step nilpotent Lie algebra need not admit an invariant metric.For example, the Heisenberg Lie algebra of any odd dimension admits no invariant metric.However, there are 2-step nilpotent Lie algebras that do admit invariant metrics.G. Ovando [52] has given some examples.For example the semidirect product of a Lie algebra with its dual carries a split signature invariant metric, and if the original Lie algebra is 2-step nilpotent, its semidirect product with its dual is 2-step nilpotent as well.However, these examples obtained from 2-step nilpotent real Lie algebras cannot be Euclidean -the invariance of the metric and the 2-step nilpotence of g imply that rg, gs is a nontrivial isotropic subspace.Ÿ Example 5.2.By Example 3.2 an antiflexible algebra has anti-self-adjoint-curvature, so a metrized antiflexible algebra has vanishing sectional nonassociativity.The following example of a three-dimensional antiflexible algebra that is not power-associative is a modification (by unimportant scalar factors) of one given by F. Kosier in [34, p. 303].Suppose char k ‰ 2 and equip k 3 with the product x ˝y " p2x 1 y 1 `x2 y 3 , 2x 1 y 2 , 2x 3 y 1 q where x " px 1 , x 2 , x 3 q and y " py 1 , y 2 , y 3 q.The two possible third powers of p0, 1, 1q are different, so the algebra is not power-associative.
(2) The quantity hpx ˝y, x ˝yq equals the Gram determinant hpx, xqhpy, yq ´hpx, yq 2 .It follows from ( 1) and the invariance of h that the product x ˝y is h-orthogonal to x and y, hpx ˝y, xq " 0. In [13] and [9] it is shown that a real cross product algebra exists only in dimensions 3 and 7, in which case it is determined uniquely up to isomorphism by the signature of h, the only possibilities being the positive definite case, and the case with negative inertial index one greater than the positive inertial index.
Theorem 5.4.Let k be an infinite field such that char k ‰ 2. A metrized algebra pA, ˝, hq with antisymmetric multiplication is a cross product algebra if and only if it has constant sectional nonassociativity 1.
Proof.The antisymmetry of ˝means that (4.13) takes the form (1.2) It follows that a cross product algebra has constant sectional nonassociativity 1.On the other hand, if a metrized algebra pA, ˝, hq with antisymmetric multiplication has constant sectional nonassociativity 1 then hpx ˝y, x ˝yq " hpx, xqhpy, yq ´hpx, yq 2 whenever x and y span an h-nondegenerate subspace.Because k is infinite, arguing as in the proof of Lemma 4.3, this holds if and only if hpx ˝y, x ˝yq " hpx, xqhpy, yq ´hpx, yq 2 for all x and y, so that pA, ˝, hq is a cross product algebra.
Corollary 5.1.Suppose char k ‰ 2. A metrized algebra pA, ˝, hq with anisotropic h satisfies hpx ˝y, x ˝yq " hpx, xqhpy, yq ´hpx, yq 2 for all x, y P A if and only if pA, ˝, hq is a cross product algebra.Proof.To show the forward direction it suffices to show ˝is antisymmetric.If hpx˝y, x˝yq " hpx, xqhpy, yqh px, yq 2 for all x, y P A, then hpx ˝x, x ˝xq " 0 for all x P A. Because h is anisotropic, this implies x ˝x " 0 for all x P A. Because char k ‰ 2, this implies ˝is antisymmetric.The reverse implication is immediate.
Hence a cross product algebra can be defined as a metrized algebra with antisymmetric multiplication and having sectional nonassociativity 1 on every h-nondegenerate subspace.Moreover, it is not necessary to suppose antisymmetry if the metric is anisotropic and char k ‰ 2. Ÿ Example 5.5.An example of a commutative algebra having constant negative sectional nonassociativity is the following.Consider R n with the coordinatewise product x ¨y " ř n i"1 x i y i .Define ℓ : R n Ñ R by ℓpxq " ř n i"1 and define a modified product x ˝y " n`1 n´1 x ¨y ´1 n´1 pℓpxqy `ℓpyqxq.Then pR n , ˝q is a commutative algebra metrized by its Killing form τ ˝px, yq " tr L ˝pxqL ˝pyq " n`1 n´1 ℓpx ¨yq ´1 n´1 ℓpxqℓpyq, (5.2) which is positive definite.With this metric, this algebra satisfies the reverse Norton inequality; in fact it has constant negative sectional nonassociativity.See [18] for details.Ÿ Example 5.6.This example describes a one-parameter family of pairwise nonisomorphic 3-dimensional Euclidean metrized commutative algebras having constant sectional nonassociativity.For ǫ P r0, 8q, define a commutative algebra pU ǫ , ˝q as the 3-dimensional R-vector space U ǫ with basis tf 0 , f 1 , f 2 u and multiplication ˝given by (5.3) shows that the metric hpx, yq " x 0 y 0 `x1 y 1 `x2 y 2 satisfies hpx ˝y, zq " x 0 y 0 z 0 `p 1 2 ´ǫqpx 1 y 1 z 0 `x1 y 0 z 1 `x0 y 1 z 1 q `p 1 2 `ǫqpx 2 y 2 z 0 `x2 y 0 z 2 `x0 y 2 z 2 q. (5.6)Because this is completely symmetric in x, y, and z it shows h is invariant with respect to ˝.From (5.4) and (5.5) it follows that the matrix of A ˝px, yq " L ˝px ˝yq ´L˝p xqL ˝pyq is ´ǫ2 q phpx, yq Id Uǫ ´x b hpy, ¨qq , (5.7)Note that tr A ˝px, yq " 2p1{4 ´ǫ2 qhpx, yq, so that h is determined by ˝.From (5.7) it follows that Kpx, yq " ´ǫ2 for all linearly independent x and y in U ǫ , so that pU ǫ , ˝, hq has constant sectional nonassociativity `p 1 2 `ǫqx 2 2 q " 1 4 tr L ˝pxq tr L ˝px ˝xq.(5.8) If I Ă U ǫ is a nontrivial ideal so is its h-orthogonal complement I K , so without loss of generality it can be assumed I is one-dimensional and generated by z P U ǫ .Consequently there is λ P k such that z ˝z " λz.Since L ˝pzqU ǫ Ă I, L ˝pzq has rank at most 1 and tr L ˝pzq " λ.By (5.8), 0 " det L ˝pzq " pλ{4qptr L ˝pzqq 2 " λz 2 0 .Were z 0 ‰ 0, then λ " 0 and z ˝z " 0. By (5.5), z ˝z " 0 if and only if p1 ´2ǫqz 1 z 0 " 0, p1 `2ǫqz 2 z 0 " 0, and z 2 0 `p1{2 ´ǫqz 2 1 `p1{2 `ǫqz 2 2 " 0. Because 2ǫ R t˘1u, if z 0 ‰ 0, the first two equations imply z 1 " 0 and z 2 " 0, which in the third equation yield the contradiction z 2 0 " 0. Consequently, z 0 " 0 and there are a, b P R such that z " af 1 `bf 2 and λaf 1 `λbf 2 " λz " z ˝z " pp 1  2 ´ǫqa 2 `p 1 2 `ǫqb 2 qf 0 .Because 2ǫ R t˘1u, were λ ‰ 0 this would imply a " 0 " b, a contradiction.If λ " 0 then p 1 2 ´ǫqa 2 `p 1 2 `ǫqb 2 " 0. If either of a or b is 0 then so is the other.Were ab ‰ 0, then it would follow from (5.4) that L ˝pzq " L ˝paf 1 `bf 2 q has rank 2, contradicting that L ˝pzq has rank at most 1.
Proof.The algebra pU 1{2 , ˝q is associative, isomorphic to a direct sum of a trivial ideal spanned by f 1 and a two-dimensional ideal spanned by f 0 and f 2 that is isomorphic to the quadratic algebra Rrts{pt 2 ´1q.When ǫ ‰ 1{2, pU ǫ , ˝q is not power associative.For example, for x " f 1 `f2 , ppx ˝xq ˝xq ˝x " p 1 2 `2ǫ 2 qf 0 while ppx ˝xq ˝px ˝xqq " f 0 .Consequently, for ǫ ‰ 1{2, pU ǫ , ˝q is not isomorphic to the associative algebra pU 1{2 , ˝q.
An algebra isomorphism maps idempotents to idempotents and preserves the spectra of the corresponding left multiplication endomorphisms.By Lemma 5.2, the algebra pU 0 , ˝q has a unique nontrivial idempotent and only one one-dimensional subalgebra, while for all ǫ ą 0, pU ǫ , ˝q has at least three distinct idempotents, so, for ǫ ą 0, pU ǫ , ˝q is not isomorphic to pU 0 , ˝qq.

Examples of bounds on sectional nonassociativity
A Euclidean metrized algebra pA, ˝, hq has nonnegative sectional nonassociativity if and only if 0 ď hprx, x, ys, yq " hpx ˝x, y ˝yq ´hpx ˝y, y ˝xq, for all x, y P A. (6.1)That is, the associator endomorphism A ˝px, xq " rx, x, ¨s is nonnegative definite for all x P A. For metrized commutative algebras (6.1) is known as the Norton inequality because S. P. Norton showed that it holds for the Griess algebra of the monster finite simple group [12,46].Example 6.1.A vertex operator algebra (VOA) over a field k of characteristic zero is a k-vector space V equipped with a linear state-field correspondence Y : V Ñ pEnd Vqrrz, z ´1ss taking values in endomorphismvalued formal power series and written Y pa, zq " ř nPZ a pnq z ´n´1 P pEnd Vqrrz, z ´1ss, and two distinguished vectors, the vacuum vector 1 P V and the conformal vector ω P V, that together satisfy some axioms that can be found in [21,22,31].
The axioms include that the endomorphisms L m " ω pm`1q satisfy rL m , L n s " pm´nqL m`n `m3 ´m 12 cδ m`n,0 for some c P k called the central charge of the VOA, and that the operator L 0 " ω p1q is semisimple and there is a direct sum decomposition V " ' 8  n"0 V n where V n is the eigenspace of L 0 with eigenvalue n, which is moreover finite-dimensional for all n ě 0. These force 1 P V 0 and V i pnq V j Ă V i`j´n´1 .A VOA is OZ (short for one-zero) if V 0 is generated by 1 and V 1 " t0u.For an OZ VOA, the multiplication ‹ defined by a ‹ b " a p1q b for a, b P V 2 makes V 2 into a commutative algebra and the symmetric bilinear form g defined by gpa, bq1 " a p3q b for a, b P V 2 is invariant with respect to ‹.This is explained in detail in [25, section 3.5] and [41,Section 8].By definition, L ‹ pωq " 2 Id A and gpω, ωq " c{2.Hence e " ω{2 is a unit in V 2 such that gpe, eq " c{8.The triple pV 2 , ‹, gq is called the Griess algebra of the OZ VOA.
By a theorem of Miyamoto [43,Theorem 6.3], the Griess algebra of a real OZ VOA having a positive definite invariant bilinear form satisfies the Norton inequality, so has nonnegative sectional nonassociativity.
Because, as mentioned in the introduction, the simple Euclidean Jordan algebras can be obtained as Griess algebras of VOAs, it follows from the theorem of Miyamoto that they have nonnegative sectional nonassociativity.Theorem 6.3 indicates a direct proof (it is not hard) and the characterization of equality.Ÿ Nonpositive sectional nonassociativity is equivalent to the reverse Norton inequality, which is (6.1) with the inequality reversed.Example 5.5 gives an example of algebras satisfying the reverse Norton inequality.Example 6.2 shows the sectional nonassociativities of symmetric composition algebras can take both signs and shows sharp upper and lower bounds for them.Example 6.2.A composition algebra is symmetric if the symmetric bilinear form hpx, yq " qpx`yq´qpxqq pyq is invariant.Basic results about symmetric composition algebras are given in [32,Chapter 34].Any Hurwitz algebra pA, ˝, q, eq has an associated para-Hurwitz algebra pA, ˛, qq that is a symmetric composition algebra.Its multiplication is the isotope of ˝determined by the canonical involution and so defined by x ˛y " x ˝ȳ.It is not unital, but the unit e of the original Hurwitz algebra is a distinguished idempotent for which L ˛peq " R ˛peq is the canonical involution of the original Hurwitz algebra.Symmetric composition algebras have been classified by A. Elduque and H. C. Myung in [14,15,16,32]; not quite all examples are given by the para-Hurwitz and forms of what are known as Okubo or Petersen algebras.In particular, a symmetric composition algebra has dimension in t1, 2, 4, 8u, it is alternative, it is nonunital if dim A ą 1, and it is commutative if and only if dim A ď 2. Lemma 6.1.Let pA, ˝, qq be a real symmetric composition algebra having associated bilinear form h. If h is positive definite, the sectional nonassociativity of pA, ˝, hq satisfies |Kpx, yq| ď 1 for all linearly independent x, y P A. There holds Kpx, yq " 1 if and only if x ˝x and y ˝y are linearly dependent, and there holds Kpx, yq " ´1 if and only if x ˝y " y ˝x.
Proof.Linearizing the relation qpx ˝yq " qpxqqpyq shows that h satisfies hpx ˝y, w ˝zq `hpw ˝y, x ˝zq " hpx, wqhpy, zq " hpy ˝x, z ˝wq `hpz ˝x, y ˝wq.(6.2) Taking z " x and w " y in (6.2) yields hpx ˝y, y ˝xq `hpx ˝x, y ˝yq " hpx, yq 2 , (6.3) while the multiplicativity of q yields 2|x ˝y| 2 " |x| the inequality by the Cauchy-Schwarz inequality, and the last equality because 2|x ˝x| 2 " |x| 4 .Equality holds if and only if x ˝x and y ˝y are linearly dependent.Lemma 6.1 implies that a para-Hurwitz algebra has sectional nonassociativities of both signs provided its dimension is at least 4. For a para-Hurwitz algebra pA, ˛, qq with underlying Hurwitz algebra pA, ˝, q, eq, there holds hpx, eq " 0 if and only if x ˛x " ´1 2 hpx, xqe and x is not a multiple of e. Consequently, by Lemma 6.1, there holds K h,˛p e, xq " ´1 if hpx, eq " 0 and K h,˛p x, yq " 1 if hpx, eq " 0, hpy, eq " 0, and x and y are linearly independent.The latter case cannot occur for a two-dimensional para-Hurwitz algebra, but does occur for para-quaternions or para-octonions.On the other hand, it follows that a two-dimensional para-Hurwitz algebra has constant sectional nonassociativity ´1.
From (6.7) it is apparent that rx, ys ‹ " ´rx, ys so that pB, ‹q is Lie-admissible, with underlying Lie algebra isomorphic to slp3, Cq with the opposite of the usual Lie bracket.The underlying symmetric product x d y " pω ´ω2 qpxy `yx `2 3 hpx, yqIq is a rescaling of the usual trace-free Jordan product.
Straightforward computations show rx, y, zs ‹ " ry, rz, xss ´hpx, yqz `hpz, yqx, so that px ‹ yq ‹ x " The real linear map σ P End R pBq defined by σpxq " ´x t is an order two isometric automorphism of pB, ‹q.Its fixed point set is A " sup3q, and so pA, ‹q is a subalgebra metrized by the Euclidean inner product hpx, yq " 1 2 trpx t y `ȳ t xq.The algebra pA, ‹, hq is the compact real form of the Okubo algebra.From (6.8) it follows that ´1 ď K A,‹,h px, yq with equality if and only if rx, ys " 0. As explained in Remark 7.4, from the Böttcher-Wenzel-Chern-do Carmo-Kobayashi inequality, Theorem 7.1, it follows that |rx, ys| 2 ď 2|x| 2 |y| 2 for x, y P sup3q, so, from (6.8) it follows that 1 ě K A,‹,h px, yq; the case of equality is characterized in Theorem 7.1.

Ÿ
The remainder of this section is devoted to the formulation and proof of Theorem 6.3, which gives sharp bounds on sectional nonassociativity of simple Euclidean Jordan algebras.
Let Mpn, hq denote the n ˆn matrices over the Hurwitz algebra h.The subspace of n ˆn Hermitian matrices over h is the subspace Hermpn, hq " tx P Mpn, hq : xt " xu comprising the fixed points of the conjugate transpose x Ñ xt .The space Hermpn, hq is a commutative algebra with unit the identity matrix, I, when equipped with the symmetrized matrix product x ‹ y " 1 2 pxy `yxq, where juxtaposition, as in the expression xy, indicates the matrix product in Hermpn, hq.The algebra pHermpn, hq, ‹q is Jordan if h is associative or h " O and n " 3 [30, Section III.1, Theorem 1].The algebra pHermp2, hq, ‹q is isomorphic to the p2 `dim hq-dimensional Jordan algebra of a quadratic form often called a spin factor; this follows from the observation that the ‹-square of a trace-free element is a multiple of the unit.
The nucleus NpA, ˝q of an algebra pA, ˝q is the set of elements a P A such that there vanish the associators ra, x, ys, rx, a, ys, and ry, a, xs for all x, y P A. It is straightforward to see that if pA, ˝q is commutative, then NpA, ˝q " ta P A : rL ˝paq, L ˝pxqs " 0 for all x P Au.If a unital Jordan algebra pA, ˝q has nondegenerate trace form tr L ˝px ˝yq and σ P S 2 A ˚is an invariant symmetric bilinear form, then there is z P NpA, ˝q such that σpx, yq " tr L ˝ppz ˝xq ˝yq for all x, y P A [33, Chapter 3, Theorem 10].Lemma 6.2.Suppose char k ffl 2n.Let h be a d-dimensional Hurwitz k-algebra and let N " n `dnpn 1q{2.The symmetric bilinear form h on Hermpn, hq defined by hpx, yq " 1 2n tptrpxyqq " 1 n tptrpx ‹ yqq is nondegenerate and invariant and satisfies tr L ‹ px, yq " N hpx, yq for all x, y P Hermpn, hq.
From the invariance of h is follows that there is z P NpHermpn, hq, ‹q such that hpx, yq " tr L ‹ ppz ‹ xq ‹ yq for all x, y P Hermpn, hq.The center of a Jordan algebra is the subset of its nucleus comprising elements that commute with all other elements.By a theorem of A. A. Albert [1], the nucleus of a simple Jordan k-algebra (char k ‰ 2) equals its center.Consequently, since pHermpn, hq, ‹q is simple, its nucleus equals its center.Because pHermpn, hq, ‹q is central simple, its center is its subalgebra, isomorphic to k, spanned by its unit [30,Chapter 5,Section 7].It follows that z is a multiple of the unit, so h is a multiple of tr L ‹ p ¨‹ ¨q.Because hpe, eq " 1 and tr L ‹ pe ‹ eq " dim Hermpn, hq " N " n `dnpn ´1q{2 where d " dim h, it follows that tr L ‹ p ¨‹ ¨q " N h.
The simple Euclidean Jordan algebras are classified; a modern exposition is [17].A simple Euclidean Jordan algebra of rank n ě 4 is isomorphic to Hermpn, hq for h an associative Hurwitz algebra and a simple Euclidean Jordan algebra of rank n " 3 is isomorphic to Hermpn, hq for h a Hurwitz algebra.The rank 2 case consists of the spin factor algebras; as mentioned these include Hermp2, hq.
In the statement of Theorem 6.3, e ij denotes the matrix with a 1 in the ij entry and 0 in other entries.
(1) If h is associative, equality holds in the lower bound of (6.11) if and only if x and y commute with respect to the matrix product; in particular, if x and x‹x are linearly independent, then Kpx, x‹xq " 0. (2) Equality holds in the upper bound of (6.11) if and only if x and y are simultaneously equivalent under AutpHermpn, hq, ‹q to scalar multiples of e 11 ´enn and e 1n `en1 .
Recall that the principal axis theorem states that every element of Hermpn, hq (where n ď 3 if h " O) is equivalent via an element of AutpHermpn, hq, ‹q to a diagonal matrix.For h P tR, C, Hu this is well known.For h " O this is [23,Theorem 5.1]; see also [17,Theorem V.2.5].
Since  (where n " 3) and in (6.15) this shows the upper bound in (6.11).The characterization of the equality case in Lemma 7.2 again yields the characterization of the equality case in the upper bound of (6.11).
The inequality (7.4) is not valid over H and O. Taking x " iI and y " jI P Mpn, Hq shows WpMpn, hqq ě 4 when h P tH, Ou.In [24,Theorem 3.1] it is shown that WpMpn, Hqq " 4 and the equality case is characterized.Nonetheless, the proof of (7.4) given in [11] for Hermpn, Rq adapts for Hermpn, hq over any real Hurwitz algebra h P tR, C, H, Ou.Over R or C, because a Hermitian matrix is unitarily diagonalizable, it suffices to prove the claim in the case one of the matrices is diagonal.This proof requires the prinicipal axis theorem, which is valid over H and O, and one has to check that the noncommutativity of H and O does not effect the rest of the argument.This works over H, but the reduction via diagonalization faces an obstacle over O, namely that, for x, y P Hermpn, hq, the inequality (7.4) is not manifestly invariant with respect to the action of AutpHermpn, hq, ‹q.Over associative h, AutpHermpn, hqq preserves the commutator of the matrix product on Mpn, hq restricted to Hermpn, hq, but when h " O, in which case AutpHermp3, Oqq is the compact form of the simple real Lie group of type F 4 [58], it is not clear to the author whether this is true.However the partial result, that (7.4) holds when x is diagonal, is true, and this suffices for the application to sectional nonassociativity in the proof of Theorem 6.3.

Lemma 7.2.
Let h be a real Hurwitz algebra.Let f px, yq " Re tr xt y on Mpn, hq.
(3) The equality |rx, ys| 2 " 2|x| 2 |y| 2 holds in (7.5)  Proof.Suppose x P Hermpn, hq is diagonal, where n " 3 if h " O. Since x is Hermitian, its entries are real.Let x i " x ii .Then rx, ys ij " x i y ij ´yij x j " px i ´xj qy ij , the last equality because x j is real, so in the center of h.Now the proof goes through as in [11]: This proves |rx, ys| 2 ď 2|x| 2 |y| 2 for diagonal x.By the principal axis theorem, any element of pHermpn, hq, ‹q is equivalent via an automorphism of pHermpn, hq, ‹q to a diagonal matrix.Since an automorphism of pHermpn, hq, ‹q preserves the trace, it is isometric, and, when h is associative it preserves the commutator of the matrix product on Mpn, hq restricted to Hermpn, hq, when h is associative to prove |rx, ys| 2 ď 2|x| 2 |y| 2 in general it suffices to prove it when x is diagonal.By Lemma 7.1, in either case |rx, ys| 2 ď 2|x| 2 |y| 2 implies (7.5).
The characterization of the equality case (3) goes through as in the proof of [11,Lemma 1].Precisely, if there is equality in (7.6), then 0 " 4 If x is nonzero, this implies y ii " 0 for 1 ď i ď n, x i `xj " 0 when y ij ‰ 0 for i ‰ j, and that at exactly two of x 1 , . . ., x n are nonzero.Suppose i ‰ j are the indices such that x i " ´xj ‰ 0. It follows that y kl " 0 if tk, lu ‰ ti, ju.Hence x and y are scalar multiples of e ii ´ejj and e ij `eji , respectively.Since AutpHermpn, hqq contains a subgroup acting as permutations on the diagonal subalgebra of Hermpn, hq, it can be assumed that i " 1 and j " n.The proof of the equality case (4) follows from (3) exactly as in the proof of (2) of Theorem 7.1.Remark 7.2.Although the same difficulty related to diagonalization occurs in the proof of Theorem 6.3 as occurs in the proof of Lemma 7.2, the end result in Theorem 6.3 is stated in terms manifestly invariant with respect to AutpHermpn, hqq, and as a result takes the same form whether or not h is associative.This observation suggests that the upper bound on sectional nonassociativity, which is closely related to the commutator bound, is the more natural bound to consider, at least from the algebraic point of view, although it could also simply reflect a technical deficiency in the proof of Lemma 7.2.Ÿ Remark 7.3.It would be useful to know whether (7.5) is true for Hermp3, Oq.This seems likely.More, generally, it would be interesting to know whether (7.5) is true for Hermpn, Oq for n ą 3.This would follow from a principal axis theorem and a characterization of the automorphism group, but it appears that neither is known.See [69] for what is known in this regard.Ÿ Remark 7.4.For a compact semisimple real Lie algebra g with Killing form B g the number Wpgq " Upg, r ¨, ¨s, ´Bg q " sup x,yPg:x^y‰0 ´Bg prx, ys, rx, ysq B g px, xqB g py, yq ´Bg px, yq 2 (7.8) is positive and finite, by the Cauchy-Schwarz inequality.Its value is a basic automorphism invariant of pg, r ¨, ¨sq.Its calculation for particular g can be viewed as a refinement of the Böttcher-Wenzel inequality.On the other hand, for some compact simple real Lie algebras Wpgq can be estimated using the Böttcher-Wenzel inequality.For supnq and sopnq, B supnq px, yq " ´2nf px, yq and B sopnq px, yq " ´pn ´2qf px, yq where f px, yq " tr xt y, so in these cases it follows from the Böttcher-Wenzel inequality, Theorem 7.1, that Wpsopnqq ď 2 n´2 , Wpsupnqq ď 1 n .(7.9)For the special case of sopnq, viewed as antisymmetric matrices, equipped with the Frobenius norm f , the essentially equivalent quantity sup Bgpa1,a1qBgpa2,a2qB h pb1,b1qB h pb2,b2q´Bgpa1,a2q 2 B h pb1,b2q 2 ď 0, (7.11)where the positivity of the denominator follows from the Cauchy-Schwarz inequality.On the other hand, by definition of Wpgq and Wphq,
Note that Lemma 7.3 does not imply that g b h has nonpositive sectional nonassociativity.In fact, for any compact simple real Lie algebra g, sop3q b g has sectional nonassociativities of both signs [18].Remark 7.5.Although this does not seem to be realized widely, for anti-Hermitian matrices the inequality (7.4) is closely related to Vinberg's results on invariant norms on compact simple Lie algebras in [64] applied in the special case of supnq.Precisely, Vinberg shows that, for an invariant norm, || ¨|| on a compact simple Lie algebra g the quantity θpxq " sup 0‰yPg ||rx,ys|| ||y|| does not depend on the choice of invariant norm, and equals the spectral norm of ad g pxq.Since, by definition, θprx, ysq ď θpxqθpyq, taking g to be a compact simple Lie algebra of matrices, e.g.antisymmetric or anti-Hermitian matrices, and taking || ¨|| to be the Frobenius norm there results |rx, ys| f ď θpxq|y| f .It suffices then to check that the spectral norm of ad g pxq is no greater than ?2|x| f .In the case g " supnq this can be shown as follows.Since any anti-Hermitian matrix is unitarily conjugate to a diagonal matrix and the spectral norms of the adjoint representations of unitarily conjugate matrices are the same, it suffices to consider the case of diagonal x P supnq.In this case the nonzero eigenvalues of ad supnq pxq have the form λ i ´λj where λ 1 , . . ., λ n are the diagonal elements of x.
f , where λ is the element of x with the greatest modulus, the spectral norm of ad supnq pxq is no greater than ?2|x| f .Ÿ

Consequences for commutative algebras of conditions on sectional nonassociativity
This section describes some general structural consequences for commutative algebras of assumptions on the sectional nonassociativity.
Consequences of nonnegative sectional nonassociativity for commutative algebras can be found in [12, section 17] and in [42], where they are applied to the study of maximal associative subalgebras of the Griess algebra.By the analogy advocated here, associative subalgebras are analogous to flat submanifolds of a nonnegatively curved Riemannian manifold.Here such results are cast in a somewhat more general setting.
An element e P A is idempotent if e˝e " e and square-zero if e˝e " 0. Let IdempA, ˝q " t0 ‰ e P A : e˝e " eu and Nil 2 pA, ˝q " t0 ‰ z P A : z ˝z " 0u. and in (4.13) this yields (8.2).In (8.2), equality cannot hold because it would imply b " 0, contrary to hypothesis.Theorem 8.1.Let pA, ˝, hq be a Euclidean metrized commutative algebra.
(2) If pA, ˝, hq has nonpositive sectional nonassociativity, either AutpA, ˝, hq is finite or pA, ˝, hq contains a trivial subalgebra of dimension 1 or 2 (the possibilities are not exclusive).
Proof.For k ě 3, let θ be a multilinear k-form on a complex vector space V.By a theorem of H. Suzuki [61,Theorem B], either the group of linear automorphisms of θ is finite or there is a nonzero vector v P V such that θpv, . . ., v, wq " 0 for all w P V. Regard µpx, y, zq " hpx ˝y, zq as a trilinear form on A b R C.An orthogonal automorphism of pA, ˝, hq preserves µ, and extends to an automorphism of A b R C. By Suzuki's theorem if AutpA, ˝, hq is not finite there exists a `ib P A b R C such that hppa `ibq ˝pa `ibq, wq " 0 for all w P A b R C. By the nondegeneracy of h, a `ib P Nil 2 pA b R C, ˝q.Note that it is not asserted that a and b are linearly independent over R. The conclusion follows from (1) of Lemma 8.1.
For a Euclidean metrized commutative algebra pA, ˝, hq, the endomorphism L ˝peq is h-self-adjoint for any e P A. If e P IdempA, ˝q, then L ˝peq preserves xey K " ty P A : hpe, yq " 0u, for if hpe, yq " 0 then hpL ˝peqy, eq " hpy, e ˝eq " hpy, eq " 0. Define the orthogonal spectrum Spec K peq " tλ P k : there is x P kteu K such that L ˝peqx " λxu, (8.4) so that Specpeq " Spec K peq Y t1u.Because L ˝peq preserves xey K it has an eigenvector in xey K .Because hpe, eq ą 0, this eigenvector is not a multiple of e, so in this case Spec K peq is not empty.Lemma 8.2.Let pA, ˝, hq be a Euclidean metrized commutative algebra of dimension n ě 2. If 0 ‰ e P IdempA, ˝q, then 4Kpe, xq ď |e| ´2 for all x P A such that x ^e ‰ 0, with equality if and only if x hpe, eq ´1hpe, xqe P kerpL ˝peq ´1 2 Id A q.
Proof.In the setting of (1), if 0 ‰ y P A is an eigenvector of L ˝peq with eigenvalue λ and orthogonal to e, then, by (8.5), λp1 ´λq " |e| 2 Kpe, yq ď b|e| 2 , so λ 2 ´λ `b|e| 2 ě 0, which forces λ to be in the indicated range.In the setting of (2), the same argument shows that λ 2 ´λ `b|e| 2 ě 0, which forces 1 ´4b|e| 2 ě 0 and forces λ to be in the indicated range.
Next there are given some results showing that nonnegative sectional nonassociativity precludes the existence of square-zero elements.Lemma 8.4.If a nontrivial metrized algebra pA, ˝, hq satisfies A ˝A " A then its multiplication is faithful.In particular, the multiplication of a metrized semisimple commutative algebra pA, ˝, hq is faithful.
Proof.If z P ker L ˝, then, by the invariance of h, 0 " hpL ˝pzqx, yq " hpz, x ˝yq for all x, y P A. This shows ker L is contained in the orthocomplement pA ˝Aq K of A ˝A.In particular, if A ˝A " A, then ker L ˝Ă pA ˝Aq K " A K " t0u, so L ˝is injective.If A " ' k i"1 A i is a direct sum of simple ideals, then A i ˝Aj " t0u if i ‰ j, and, since A i is simple, A i ˝Ai " A i , so A ˝A " ř i A i ˝Ai " ř i A i " A, and L ˝is injective.By the invariance of h, hpR ˝pyqx, zq " hpx, L ˝pyqzq for all x, y, z P A, so were R ˝not injective, L would not be injective.
Lemma 8.5.Let pA, ˝, hq be a Euclidean metrized commutative algebra.Let 0 ‰ z P Nil 2 pA, ˝q.For 0 ‰ y P A not in the span of z, Kpz, yq ď 0, with equality if and only if z ˝y " 0. In particular, Nil 2 pA, ˝q " t0u if there holds either of the following conditions: (1) pA, ˝, hq has positive sectional nonassociativity.
Proof.Let 0 ‰ z P Nil 2 pA, ˝q.For y P A linearly independent of z, there holds Kpz, yq|z ^y| 2 " ´|L ˝pzqy| 2 ď 0, with equality if and only if z ˝y " 0. In particular, pA, ˝, hq cannot have positive sectional nonassociativity, and if pA, ˝, hq has nonnegative sectional nonassociativity, then L ˝pzqy " 0 for all y P A, so L ˝pzq " 0. By Lemma 8.4, if L ˝pzq " 0, then pA, ˝q is not semisimple.
A commutative algebra is exact if tr L ˝pxq " 0 for all x P A. Exactness is a condition analogous to unimodularity of a Lie algebra.Evidently, an exact commutative algebra is not unital.Lemma 8.6.A Euclidean metrized commutative algebra pA, ˝, hq with nontrivial multiplication and nonnegative sectional nonassociativity is not exact.
In a commutative R-algebra pA, ˝q the set QpA, ˝q " tx ˝x : x P Au is a closed cone, the cone of squares.By definition, IdempA, ˝q Ă QpA, ˝q.Lemma 8.7.Let pA, ˝, hq be a Euclidean metrized commutative algebra having nonnegative sectional nonassociativity.The set CpA, ˝, hq " tx P A : L ˝pxq is nonnegativeu (8.7) is a closed convex cone containing QpA, ˝q for all x P A and consequently containing IdempA, ˝q.
Proof.That the sectional nonassociativity is nonnegative means that L ˝px ˝xq ě L ˝pxq 2 ě 0 for all x P A. This shows IdempA, ˝q Ă QpA, ˝, hq Ă CpA, ˝, gq.In a Euclidean metrized commutative algebra every multiplication endomorphism L ˝pxq is self-adjoint, so diagonalizable with real eigenvalues.If x, y P A and t P r0, 1s then L ˝ptx `p1 ´tqyq " tL ˝pxq `p1 ´tqL ˝pyq ě 0 because the set of nonnegative definite self-adjoint endomorphisms of a Euclidean vector space is a closed convex cone.
With the hypotheses of Lemma 8.7, are the extreme rays of QpA, ˝q generated by elements of IdempA, ˝q?This is true for the cones of squares in the simple Euclidean Jordan algebras over R or C; these are the cones of nonnegative definite Hermitian matrices and their extremal rays are generated by the rank one idempotents.For the Griess algebras of OZ VOAs (Example 6.1) it seems the question has not been studied.

Example 3 . 7 .Lemma 3 . 4 .
Commutative algebras and Lie algebras are associator-cyclic, but a general anticommutative algebra need not be associator-cyclic.Ÿ If char k ‰ 3, an associator-cyclic k-algebra has self-adjoint curvature if and only if it is flexible.Proof.By Lemma 3.1, a flexible algebra has self-adjoint curvature, so it suffices to prove that an associatorcyclic algebra with self-adjoint curvature is flexible.Taking x " z in (3.8) and (3.4) yields rx, y, xs " ´rx, x, ys ´ry, x, xs " ´2rx, y, xs, so that 3rx, y, xs " 0 for all x, y P A. If char k ‰ 3, this implies pA, ˝q is flexible.Antisymmetrizing (3.8) in x and y shows that an associator-cyclic algebra is Lie-admissible.Lemma 3.5 shows that these conditions are equivalent for flexible k-algebras when char k ‰ 2. That when char k ‰ 2 an algebra is flexible and Lie-admissible if and only if there holds (2) of Lemma 3.5 is due to Okubo and Myung in[44, p. 80] and [50, Theorem 4.1].Lemma 3.5 ([44, p. 80], [50, Theorem 4.1],

Lemma 7 . 3 .
x,yPg:x^y‰0 |rx,ys| f |x| f |y| f was shown to equal ? 2 if n ě 4 in [5, Theorem 6].By Lemma 7.3, these estimates yield sharp numerical bounds on the sectional nonassociativites of the subspaces spanned by decomposable elements in algebras such as sopmq b sopnq, sopmq b supnq, and supmq b supnq.Ÿ Lemma 7.3 shows a partial lower bound on the sectional nonassociativities of the tensor product of Lie algebras.For compact semisimple real Lie algebras g and h, let τ ˝be the Killing form of the tensor product algebra pg b h, ˝q, defined by τ ˝pa, bq " tr L ˝paqL ˝pbq for a, b P g b h.The sectional nonassociativity of the subspace of pg b h, ˝, τ ˝q spanned by decomposable elements a 1 b b 1 , a 2 b b 2 P g b h satisfies 0 ě Kpa 1 b b 1 , a 2 b b 2 q ě ´WpgqWphq.(7.10) Proof.Because g and h are compact, their Killing forms, B g and B h , satisfy ´Bg pra 1 , a 2 s, ra 1 , a 2 sq ě 0 and ´Bh prb 1 , b 2 s, rb 1 , b 2 sq ě 0. The upper bound in (7.10) follows from Kpa 1 b b 1 , a 2 b b 2 q " ´Bgpra1,a2s,ra1,a2sqBhprb1,b2s,rb1,b2sq

Lemma 8 . 1 .
Let pA, ˝, hq be a Euclidean metrized commutative algebra.(1) If a `ib P Nil 2 pA b R C, ˝q and the R-span of a and b is two-dimensional, then Kpa, bq " |a˝a| 2 |a^b| 2 ě 0, (8.1) with equality if and only if a and b span a trivial subalgebra of pA, ˝q.(2) If a `ib P IdempA b R C, ˝q and the R-span of a and b is two-dimensional, then Kpa, bq " That a `ib P Nil 2 pA b R C, ˝q is equivalent to the equations a ˝a " b ˝b and a ˝b " 0, and in (4.13) these yield (8.1).If equality holds in (8.1), then b ˝b " a ˝a " 0, so a and b generate a trivial subalgebra.Suppose a `ib P IdempA b R C, ˝q, so that a ˝a ´b ˝b " a and 2a ˝b " b.Then hpa ˝a, b ˝bq ´hpa ˝b, a ˝bq " hpb ˝b `a, b ˝bq " |b ˝b| 2 `hpb, a ˝bq ´1 4 |b| 2 " |b ˝b| 2 `1 4 |b| 2 , (8.3) takes the form Kpx, yq " hprx, ys, rx, ysq hpx, xqhpy, yq ´hpx, yq 2 .(1.2) Example 2.1.A semisimple real Lie algebra pg, ˝q is metrized by the negative h " ´Bg of its Killing form B g " tr L ˝pxqL ˝pyq.It is Euclidean in the above sense if it is moreover compact.More generally, an anticommutative algebra pA, ˝q is Malcev if px ˝yq ˝px ˝zq " ppx ˝yq ˝zq ˝x `ppy ˝zq ˝xq ˝x `ppz ˝xq ˝xq ˝y, Any Lie algebra is Malcev.By [56, Theorem 7.16] the Killing form τ ˝px, yq " tr L ˝pxqL ˝pyq of a Malcev algebra is invariant.The identity (2.1) can be written L ˝px ˝yqL ˝pxq `L˝p xqL ˝px ˝yq " rL ˝pxq 2 , L ˝pyqs and tracing this shows τ ˝px ˝y, xq " τ ˝px, y ˝xq.Polarizing this in x yields the claim.
for all x, y, z P A.(2.1)for all x, y, z P A. Ÿ Lemma 2.1 generalizes the statement that a Lie algebra admitting an invariant nondegenerate bilinear form has dimension at least 3.Lemma 2.1.A two-dimensional metrized algebra pA, ˝, hq is commutative.
An algebra is left-symmetric (also called pre-Lie or Vinberg) if A ˝px, yq " A ˝py, xq for all x, y P A. It is evident from (3.1) that an algebra is left-symmetric if and only if its curvature r vanishes.The vanishing of the adjoint curvature is equivalent to the left symmetry of the adjoint multiplication, or that the given multiplication is right-symmetric.
If either x, y P Hermpn, Cq or x, y P supnq, then |rx, ys| 2 " 2|x| 2 |y| 2 if and only if x and y are simultaneously unitarily conjugate to scalar multiples of e 11 ´enn and e 1n `en1 .(2) If either x, y P Hermpn, Cq or x, y P supnq, then |rx, ys| 2 " 2p|x| 2 |y| 2 ´f px, yq 2 q if and only if x and y either are linearly dependent over C or are simultaneously unitarily conjugate to scalar multiples of e 11 ´enn and e 1n `en1 .
(where, if h " O and n " 3, x is assumed diagonal) if and only if x and y are simultaneously equivalent via an automorphism of the Jordan algebra pHermpn, hq, ‹q to real multiples of e 11 ´enn and e 1n `en1 .(4) The equality |rx, ys| 2 " 2p|x| 2 |y| 2 ´f px, yq 2 q holds in (7.5) (where, if h " O and n " 3, x is assumed diagonal) if and only if x and y are either linearly dependent over h or are simultaneously equivalent via an automorphism of the Jordan algebra pHermpn, hq, ‹q to real multiples of e 11 ´enn and e 1n `en1 .