Background
Vector-valued cross products
Let $\mathbb{F}$ be an field of characteristic $0$. A $k$-ary cross product in the vector space $V=\mathbb{F}^n$ equipped with an inner product $\langle \cdot, \cdot \rangle$ is a $k$-linear map $\times: V^k \to V$ which outputs a vector with the following two properties:
It is orthogonal to all inputs.
Its norm is the volume of the $k$-dimensional paralellotope formed by these inputs, that is,
$$|\!\times\!\!(\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_k)|^2 = \det \langle \mathbf{v}_i, \mathbf{v}_j \rangle,$$
where the right hand side is a Gram determinant. These two conditions turn out to imply that composition with the inner product $\langle \times, \cdot \rangle$ ("lowering an index") defines a totally antisymmetric $(k+1)$-linear map $\times : V^{k+1} \to \mathbb{F}$, i.e. an alternating $(k+1)$-form. Equivalently, given a $(k+1)$-form satisfying the appropriate volume condition one can define a cross product by "raising an index".
In the following we restrict to $\mathbb{F}=\mathbb{R}$ with the standard inner product. The dimensions where a cross product exists have been completely classified (see for example here):
There is a trivial $k$-ary cross product for any $k\ge n+1$, which is identically zero.
A nullary cross product (equivalently a unit vector) exists for any $n\ge 1$.
A unary cross product (corresponding to a symplectic form) exists in any even dimension $n=2m$.
A $(n-1)$-ary cross product (corresponding to a volume form) exists in any dimension $n$. For example, the ordinary cross product in 3D is of this kind.
There are only two exceptional cross products not covered by the above cases, binary in dimension $7$ (corresponding to an associative $3$-form) and ternary in dimension $8$ (corresponding to a Cayley $4$-form). They can both be expressed in terms of the octonions, see for example this question for details.
So we have the following table:
$$\begin{array} {|c|c|c|c|c|} \hline k\setminus n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & \cdots\\ \hline 0 & & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \ldots\\ \hline 1 & \checkmark & & \checkmark & & \checkmark & & \checkmark & & \checkmark & & \checkmark & & \ldots\\ \hline 2 & \checkmark & \checkmark & & \checkmark & & & & \checkmark & & & & &\\ \hline 3 & \checkmark & \checkmark & \checkmark & & \checkmark & & & & \checkmark & & & &\\ \hline 4 & \checkmark & \checkmark & \checkmark & \checkmark & & \checkmark & & & & & & &\\ \hline 5 & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & &\checkmark & & & & & &\\ \hline 6 & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & &\checkmark & & & & &\\ \hline \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & & \ddots & & & &\\ \hline\end{array}$$
Note that if there is a $k$-ary product in dimension $n$, there is automatically a $(k-1)$-ary product in dimension $n-1$ obtained by contracting with an unit vector. This explains the diagonal patterns.
Bivector-valued cross products
Similarly, we can define a $k$-ary bivector-valued cross product (BV cross product or BVCP for short), which is a $k$-linear map $\times : V^k \to \Lambda^2 V$ outputting a bivector with the following two properties:
It is orthogonal to all inputs (i.e. the left contraction $\mathbf{v}_j \: \lrcorner \times\!\!(\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_k)$ vanishes for all $1\le j \le k$).
Its norm is the volume of the paralellotope formed by these inputs.
As before, given any BV cross product we may define a corresponding $(k+2)$-form and viceversa, by lowering or raising two indices. Note that the wedge/outer product of two vectors is not a BV cross product, since it doesn't satisfy the first condition (in fact it does the complete opposite, since the plane spanned by $a \wedge b$ contains $a$ and $b$).
So far I haven't been able to find a classification of BV cross products anywhere (this might be because I've been searching for the wrong terms). However, we can give a partial classification by trying to find bivector-valued analogues of each class of vector-valued cross product:
There are trivial $k$-ary BV cross products for any $k\ge n+1$.
A nullary BV cross product, or equivalently a bivector of norm $1$, exists for any $n\ge 2$.
A unary BV cross product exists in any dimension $n=3a+7b$, since then we can decompose $V$ into the direct sum of vector spaces of the form $\mathbb{R}^3$ and $\mathbb{R}^7$, and take a corresponding $3$-form which restricts to a suitable multiple of the volume form in each $\mathbb{R}^3$ and a multiple of the associative $3$-form in each $\mathbb{R}^7$. On the other hand, no unary product exists for $n = 1, 2$ for dimensional reasons. Since every number greater than $11$ can be expressed as $3a+7b$ for some $a, b$, this leaves in doubt only the cases $n=4, 5, 8$ and $11$. (The cases $n=4, 5$ are now ruled out and $n=8, 11$ are confirmed, see the updates).
A $(n-2)$-ary cross product (corresponding to a properly normalized volume form) exists in any dimension $n$.
The Hodge dual of the associative $3$-form, properly normalized, induces a binary BV cross product in dimension $7$, and similarly the Cayley form itself induces another in dimension $8$. I'm sure there is a better way to prove it, but I just used the well-known correspondence between bivectors and skew-symmetric matrices and used a CAS to check symbolically that the relations $M_{\mathbf{x}\times\mathbf{y}}\mathbf{x} = M_{\mathbf{x}\times\mathbf{y}}\mathbf{y} = 0$ and $||M_{\mathbf{x}\times\mathbf{y}}|| = |\mathbf{x}\wedge\mathbf{y}|$ hold, where $||\cdot||$ is the Frobenius norm.
Here is a tentative table summarizing the above:
$$\begin{array} {|c|c|c|c|c|} \hline k\setminus n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & \cdots\\ \hline 0 & \color{darkred}\bullet & \color{darkred}\bullet & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \ldots\\ \hline 1 & \checkmark & & & \checkmark & \color{darkred}\bullet & \color{darkred}\bullet & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \ldots\\ \hline 2 & \checkmark & \checkmark & & & \checkmark & & & \checkmark & \checkmark & & & &\\ \hline 3 & \checkmark & \checkmark & \checkmark & & & \checkmark & & & \color{darkred}\bullet & \color{darkred}\bullet & & &\\ \hline 4 & \checkmark & \checkmark & \checkmark & \checkmark & & & \checkmark & & & & & &\\ \hline 5 & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & & &\checkmark & & & & &\\ \hline 6 & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & & &\checkmark & & & &\\ \hline \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & & & \ddots & & &\\ \hline\end{array}$$
Question
My question is:
Is the list above correct? Are there any other possible BV cross products, e.g. binary in dimension $10$?
UPDATE:
We can generalize the Hodge dual argument we used to get the $7$-dimensional binary BVCP to rule out the existence of BVCPs for some $n$ and $k$.
First, note that if we know a cross product doesn't exist for some $n$ and $k$, automatically we know that none exists for $n+a$ and $k+a$ with any $a>0$, since otherwise we could contract the latter with $a$ unit vectors and obtain a contradiction.
A $p$-vector-valued cross product is the obvious generalization of vector-valued ($p=1$) and bivector-valued ($p=2$) cross products to multivectors of arbitrary degree. With some effort, I have proven that a $(n,k,p)$-cross product induces a $(n,k,n-2k-p)$-cross product by lowering $k$ indices, taking the Hodge dual, raising $k$ indices again and renormalizing. Conversely, if there are no $(n,k,p)$-cross products, we can deduce the nonexistence of $(n,k,n-2k-p)$-cross products.
Note that for scalar-valued cross products ($p=0$) the orthogonality condition is trivially satisfied. Their classification is easy to state: they exist only for $k\ge n$ or for $k=0$, $n$ arbitrary.
Using the classifications of scalar and vector-valued products and applying Hodge duality to $(n,k,p) = (4,1,0)$, $(5,1,1)$, $(8,3,0)$ and $(9,3,1)$, we prove the nonexistence of BVCPs at $(4,1,2)$, $(5,1,2)$, $(8,3,2)$ and $(9,3,2)$, and of any others as we move diagonally in the bottom right direction.
I marked some cells in the table with dark-red dots where we know BVCPs don't exist. Note that the diagonal below any dot is also ruled out by the discussion above.
In light of this new information, the question can now be divided in two main subquestions:
Are there unary BVCPs in dimensions $8$ and $11$?
UPDATE 2: I have found an $8$-dimensional unary BVCP satisfying the conditions, whose corresponding $3$-form is given by a rescaling of the structure constants of the Lie algebra $\mathfrak{su}(3)$. It follows that an unary product exists in dimensions $n=3a+7b+8c$, and in particular also for $n=11$. I updated the table with these two new cases.
- Are there binary BVCPs in dimension greater than $8$?
