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 $x,y,z\in R$.

Equivalence of definitions

Further information: equivalence of definitions of 2-Engel alternating ring