Division algebras satisfying $(x^p, x^q, x^r)=0$

Abstract

We study algebras $A,$ over a field of characteristic zero, satisfying $(x^p, x^q, x^r)=0$ for $p, q, r$ in ${1, 2}.$ The existence of a unit element in such algebras leads to the third power-associativity. If, in addition, $A$ has degree $\leq 4$ then $A$ is power-commutative. We deduce that any 4-dimensional real division algebra, with unit element, satisfying $(x^p, x^q, x^r)=0$ is quadratic. This persists for $(x, x^q, x^r)=0$ if we replace the word "unit" by "left-unit".