I've always heard that the cross product only exists in a well defined way for 3 and 7 dimensions. From my own reading I've found that a cross product in 3 dimensions is nothing more than the product of two quaternions with only pure imaginary components (that is, the real part is zero). Likewise, a seven dimensional cross product is simply the product of two octonions where again the real part is zero. But why can't this process simply continue? Why not construct the 16 component sedenians and define the 15 dimensional cross product to be the product of two sedenians with zero real parts? Is there some property of sedenians that disallows this?