Magma in which cubes are well-defined and every element commutes with its cube

Definition
A magma in which cubes are well-defined and every element commutes with its cube is a magma $$(S,*)$$ satisfying the following two conditions:


 * 1) For every $$a \in S$$, $$a$$ commutes with the value $$a^2 = a * a$$. In other words, $$a * a^2 = a^2 * a$$. This common value is denoted $$a^3$$.
 * 2) For every $$a \in S$$, $$a * a^3 = a^3 * a$$.