On the solvability of commutative power-associative nilalgebras of nilindex 4

Abstract

Let $A$ be a commutative power-associative nilalgebra. In this paper we prove that when $A$ (of characteristic $\neq 2)$ is of dimension $\leq 10$ and the identity $x^{4}=0$ is valid in $A$, then $((y^{2})x^{2})x^{2}=0$ for all $y$, $x$ in $A$ and $((A^{2})^{2})^{2}=0$. That is, $A$ is solvable.