This is a theorem that under approximately those rules* in R^n you cannot have inverses except in the cases of the reals, complexes, and quaternions: https://en

This is a theorem that under approximately those rules* in R^n you cannot have inverses except in the cases of the reals, complexes, and quaternions: https://en.wikipedia.org/wiki/Frobenius_theorem_%28real_division_algebras%29 https://en.wikipedia.org/wiki/Frobenius_theorem_%28real_divi... *It doesn't rely on rule 2, it does however rely on (u*v)*w = u*(v*w). Adding rule 2 excludes the quaternions.