Half-axes in power associative algebras

We show that if identities (⁎) hold strictly in A, then one gets (very) interesting identities between elements in the eigenspaces of ade (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.