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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?

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    $\begingroup$ One ought to say what one means by cross product in the first place. One definition is in this classic article of Brown and Gray: link.springer.com/article/10.1007%2FBF02564418 pre-kidney's answer is consistent with the definition of Eckmann given at the beginning of the article (and restricting to the case of cross products with two arguments). $\endgroup$ – Travis Willse Jan 17 '16 at 8:40
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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).

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