Magma in which cubes and fourth powers are well-defined

Definition
A magma $$(S,*)$$ is termed a magma in which cubes and fourth powers are well-defined if it satisfies the following conditions for all $$a \in S$$ (here, $$a^2$$ is shorthand for $$a * a$$):


 * 1) Cubes are well-defined: $$a^2 * a = a * a^2$$, and the common value is denoted $$a^3$$.
 * 2) Fourth powers are well-defined: $$a^3 * a = a^2 * a^2 = a * a^3$$, and the common value is denoted $$a^4$$.