Jordan Identity over $char\neq2$ implies power-associativity?

Question

Let $A$ be non-comutative Algebra over a field of characteristic not 2 that satisfies $(xx)(xy)=x(y(xx))$. Can we say that $A$ is power-associative?

My attempt: I'm trying to disprove the claim for any field but only found a counterexample for characteristic 2. I do know that a Jordan Algebra is power-associative but the comutativeness seems crucial therefore I believe the answer for the above question is "no". Any example is welcome.

Answer 1

Yes, Schafer proved in $1955$ that any noncommutative Jordan algebra of characteristic different from $2$ is power-associative. For the proof see here.