  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.
