Malcev ring

Definition
A Malcev ring (also called a Moufang-Lie ring) is a non-associative ring (i.e., a not necessarily associative ring) $$M$$ with multiplication $$*$$ that satisfies the following two conditions:


 * The ring is an alternating ring: $$\! x * x= 0 \ \forall \ x \in M$$
 * The multiplication satisfies the Malcev identity, i.e., the following is true for all $$x,y,z \in M$$:

$$\! (x * y) * (x * z) = (((x * y) * z) * x) + (((y * z) * x) * x) + (((z * x) * x) * y)$$