Higher dimensional cross product

Question

I know that cross products do not exist in 4, 5 or 6 dimensions, but do in 7 dimensions. So I was wondering if this was because cross products can be considered the imaginary part of $2^n - ion$ product. I tried it out for 3 dimensional cross product (as imaginary part of quaternion product) and did not run into any troubles or contradictions. Could this hold for octonions as well? My concern is their multiplicative non-associativity. If my assumption is correct, then what about sedonions and higher $2^n-ion$s?

Answer 1

First of all, note that the usual cross product already isn't associative! The seven-dimensional cross product also arises in this manner from the octonions.

As pointed out in the latter link, the only dimensions in which cross products exist is $0,1,3,7$. This is because normed division algebras only exist in dimensions $1,2,4,8$: even though there are a family of $2^n$ dimensional Clifford algebras which generalize the quaternions, octonions, sedonions, etc. in the way you are probably imagining, the sedonions are not alternative: and this is the straw that "breaks the camel's back", and prevents there from being a $15$-dimensional cross product (as well as all higher $2^n-1$ dimenional analogues).