A uniform characterization of the octonions and the quaternions using commutators

TL;DR

This paper characterizes octonions and quaternions through properties of rings with non-zero commutators, showing under certain conditions such rings are division algebras of these types.

Contribution

It provides a unified, elementary proof that rings with specific commutator properties are either octonion or quaternion division algebras.

Findings

Rings with non-zero commutators are free of zero divisors.

Localization yields octonion or quaternion division algebras depending on associativity.

The proof is elementary and applies to both alternative and associative cases.

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.