Half-axes in power associative algebras

Abstract

Let $A$ be a commutative, non-associative algebra over a field $\mathbb{F}$ of characteristic $\ne 2$. A half-axis in $A$ is an idempotent $e\in A$ such that $e$ satisfies the Peirce multiplication rules in a Jordan algebra, and, in addition, the $1$-eigenspace of ${\rm ad}_e$ (multiplication by $e$) is one dimensional. In this paper we consider the identities $(*)$ $x^2x^2=x^4$ and $x^3x^2=xx^4.$ We show that if identities $(*)$ hold strictly in $A,$ then one gets (very) interesting identities between elements in the eigenspaces of ${\rm ad}_e$ (note that if $|\mathbb{F}|>3$ and the identities $(*)$ hold in $A,$ then they hold strictly in $A$). Furthermore we prove that if $A$ is a primitive axial algebra of Jordan type half (i.e., $A$ is generated by half-axes), and the identities $(*)$ hold strictly in $A,$ then $A$ is a Jordan algebra.