It is a well-known mathematical curiosity that ordinary (vector-valued) cross products over $\mathbb{R}$ exist only in dimensions $0, 1, 3$ and $7$ (this fact is related to Hurwitz's theorem that real composition algebras only exist in dimensions $1, 2, 4, 8$). I am interested in a generalization of this result to bivector-valued cross products.
A bivector-valued cross product (see here for the terminology) is a bilinear map $\times : V^2 \to \Lambda^2 V$ with the following two properties:
The resulting bivector is orthogonal to all inputs (i.e. the left contractions $\mathbf{x} \, \lrcorner\, (\mathbf{x}\times \mathbf{y})$ and $\mathbf{y} \, \lrcorner\, (\mathbf{x}\times \mathbf{y})$ vanish). If we write the bivector as a skew-symmetric matrix $M_{\mathbf{x},\mathbf{y}}$, this condition is simply $M_{\mathbf{x},\mathbf{y}} \mathbf{x} = M_{\mathbf{x},\mathbf{y}} \mathbf{y} = 0$.
In analogy with ordinary cross products, the resulting bivector's norm (the Frobenius norm of the previous matrix) equals the area of the paralellogram formed by the inputs. That is, we have $$|\mathbf{x}\times \mathbf{y}|^2 = \det \begin{pmatrix} \mathbf{x}\cdot\mathbf{x} & \mathbf{x}\cdot\mathbf{y}\\ \mathbf{y}\cdot\mathbf{x} & \mathbf{y}\cdot\mathbf{y} \end{pmatrix}.$$
The following dimensions are known to admit a bivector-valued cross product:
In dimensions $0$ and $1$ the identically zero product trivially satisfies both conditions.
In dimension $4$, we can take the four-dimensional volume form, raise two indices and multiply by the appropriate normalization constant.
Finally, in dimensions $7$ and $8$ bivector-valued cross products can be defined in terms of a coassociative form (the Hodge dual of an associative form) and a Cayley form respectively, by raising two indices and normalizing.
On the other hand:
In dimensions $2$ and $3$ such a product is impossible: there is enough room that some parallelograms are nontrivial, but not enough room to fit in a nonzero orthogonal bivector.
In dimensions $5$ and $6$ there aren't any bivector-valued cross products either, though the proof I have is less straightforward (it involves Hodge duality and appealing to the known classifications of scalar-valued and vector-valued $k$-ary products, as I mention in the MSE question below).
My question is:
Does there exist any bivector-valued cross product in dimension $9$ or higher?
Note: this is the last remaining case of a more general question I asked in Math StackExchange. My hope is that there is already a reference somewhere dealing with this problem, or with something equivalent to it.
Update (02/01/2021):
An equivalent way to state the defining conditions is in terms of a totally antisymmetric tensor $f_{abcd}$ (in abstract index notation) of rank $4$ such that $(x \times y)_{ab} = f_{abcd} x^c y^d$. This tensor satisfies
$$f_{ab}{}^{ef}f_{cdfe}+f_{da}{}^{ef}f_{bcfe}=2g_{ac}g_{bd}-g_{ab}g_{cd}-g_{ad}g_{bc},$$
where $g_{ab}$ denotes the standard inner product, which I also implicitly used to raise and lower indices. This identity comes from polarizing the area condition. Note that the orthogonality condition follows from the full antisymmetry of $f$.
A related problem has been studied in Chapter 20 of the book Group Theory by P. Cvitanović. Here $f$ and $g$ (of the same rank and symmetry) are assumed primitive invariant tensors, meaning roughly that every tensor invariant under $G = \operatorname{Aut}(V,f,g)$ can be decomposed as a linear combination of certain contractions of $f$ and $g$, namely those that are tree-like in Penrose's graphical notation.${}^{(*)}$
This primitiveness condition turns out to imply that $f$ is (almost always) a bivector-valued cross product: adding to Equation (20.6) its $90{}^\circ$-rotated version and using antisymmetry of $f$, we obtain precisely the polarized area condition modulo relabeling and raising/lowering indices, except that there is an overall factor of $2b(b-1)$ multiplying the right hand side, where $b$ is a nonnegative constant. This factor can be absorbed by redefining $f\mapsto f/\sqrt{2b(b-1)}$ provided $b>1$.
The author classifies the possible Lie groups $G$ arising in this way (or rather their associated Lie algebras), as well as the corresponding dimension $n$ of the defining representation, by means of Diophantine conditions. He obtains the known nontrivial cross products in dimensions $4, 7$ and $8$, acted on by $SO(4), G_2$ and $\mathrm{Spin}(7)$ respectively (the case $n=10$ acted on by $SO(10)$ has $b=1$, so it does not produce a cross product). Thus we don't find any new examples of bivector valued cross products.
If the reverse implication somehow held (being a cross product implies being primitive), that would constitute a proof of the nonexistence of bivector valued cross products in dimension $9$ or higher. If I understand correctly, primitiveness can fail in two ways:
The tensor $f_{abcd}$ can fail to be primitive itself, in which case it is a linear combination of tree contractions of primitive invariant tensors of lower rank. I'm not entirely sure, but I think these should be able to be ruled out manually. For example, if $f_{abcd} = f'_{[ab} f'_{cd]}$ for some antisymmetric primitive $f'$ of rank $2$, a short graphical manipulation implies that there is no way to satisfy the area condition unless $n=-4$, which is impossible (but see the remark below). The case of rank $3$ seems more involved, and I haven't been able to rule it out yet.
There may exist more primitive invariant tensors. The group preserving them would presumably have to be a Lie subgroup of $SO(4), G_2$ or $\mathrm{Spin}(7)$, by the argument in Section 4.7 of the book. I'm not sure of the details, but in principle one only needs to consider the finitely many types of extra primitive invariants that would spoil the derivation of the Diophantine conditions on the dimension.
I don't really know how plausible this whole approach is, but a priori it looks like a viable way to attack the problem.
$(*)$ This sort of graphical calculus, known under many other names (birdtracks, string diagrams, trace diagrams, etc.) is very useful for manipulating tensor equations with many indices. Another reference where these diagrams are extensively applied is L. Cadorin's thesis, where it is used to classify, among other things, ternary vector valued cross products; in particular, Equation (27.10) is the same as the area condition up to a factor of $d-2$ multiplying the right hand side (note she uses a different graphical convention for antisymmetric tensors of even rank). As above, one can absorb this factor into $f$ provided the dimension $d$ is not $2$. What this tells us is that given a ternary vector valued cross product ($d=0, 1, 2, 4, 8$) in any dimension other than $2$, we obtain a corresponding binary bivector valued cross product. Again we obtain no new examples.
Interestingly, both Cadorin and Cvitanović obtain negative even solutions for the dimension, which may be interpreted as evidence for some sort of cross product-like structures in Grassmann coordinates. As Cvitanović points out, his negative solutions contain the third row of Freudenthal's magic square.
