There is actually a 7-dimensional "cross-product" (-esque operation): https://en.wikipedia.org/?title=Seven-dimensional_cross_product https://en.wikipedia.org/?

There is actually a 7-dimensional "cross-product" (-esque operation): https://en.wikipedia.org/?title=Seven-dimensional_cross_product https://en.wikipedia.org/?title=Seven-dimensional_cross_prod... The relationship between Quaternions and the 3-dimensional cross-product corresponds to the relationship between the octonions and this 7-dimensional cross-product. Normed Division Algebras are a fascinating and worthwhile subject to look into.