Half-axes in power associative algebras

Let A be a commutative, non-associative algebra over a field F of characteristic ≠2. A half-axis in A is an idempotent e∈A such that e satisfies the Peirce multiplication rules in a Jordan algebra, and, in addition, the 1-eigenspace of ad_e (multiplication by e) is one dimensional. In this paper we consider the identities (⁎) x²x²=x⁴ and x³x²=xx⁴. We show that if identities (⁎) hold strictly in A, then one gets (very) interesting identities between elements in the eigenspaces of ad_e (note that if |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.