2-Engel alternating loop ring

Definition
Suppose $$R$$ is an alternating loop ring with binary operation $$*$$. We say that $$R$$ is 2-Engel if it satisfies the following equivalent conditions:


 * 1) The subring generated by any subset of size at most two is a Lie ring of nilpotency class two.
 * 2) The additive loop of $$R$$ is diassociative and $$x * (x * y) = 0$$ for all $$x,y \in R$$.
 * 3) The additive loop of $$R$$ is diassociative and $$x * (y * z) = y * (z * x) = z * (x * y)$$ for all $$x,y,z \in R$$.