A uniform characterization of the Octonions and the Quaternions using commutators

Abstract

Let $R$ be a ring which is not commutative. Assume that either $R$ is alternative, but not associative, or $R$ is associative and any commutator $v\in R$ satisfies: $v^2$ is in the center of $R.$ We show (using commutators) that if $R$ contains no divisors of zero and $\text{char}(R)\ne 2,$ then $R//C,$ the localization of $R$ at its center $C,$ is the octonions in the first case and the quaternions, in latter case. Our proof in both cases is essentially the same and it is elementary and rather self contained. We also give a short (uniform) proof that if a non-zero commutator in $R$ is not a zero divisor (with mild additional hypothesis when $R$ is alternative, but not associative (e.g.~that $(R,+)$ contains no $3$-torsion), then $R$ contains no divisors of zero.