A uniform characterization of the octonions and the quaternions using commutators

Abstract

Let $R$ be a ring with ${\bf 1}$ which is not commutative. Assume that a non-zero commutator in $R$ is not a zero divisor. Assume further 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 prove that $R$ has no zero divisors. Furthermore, if $\text{char}(R)\ne 2,$ then the localization of $R$ at its center is an octonion division algebra, if $R$ is alternative and a quaternion division algebra, if $R$ is associative. Our proof in both cases is essentially the same and it is elementary and rather self contained.