Given an involution *, does xx* = x*x? If not, what is a counterexample?

Question

Let A be an algebra with involution. An involution $*$ is a unary operation that satisfies the following properties:

1. $\forall a \in A, (a^*)^* = a$
2. $\forall a, b \in A, (a + b)^* = a^* + b^*$
3. $\forall a, b \in A, (ab)^* = b^*a^*$

My question is then: Does the identity $\forall x \in A, xx^* = x^*x$ hold for any algebra with involution?

If so, what is the proof?

If not, are there conditions under which it does hold (besides the obvious case of a commutative algebra)? And is a counterexample in which it does not hold?