Lie ring of 2-local nilpotency class three

Definition
A Lie ring $$L$$ is termed a Lie ring of 2-local nilpotency class three if it satisfies the following equivalent conditions:


 * 1) Its 2-local nilpotency class is at most three. In other words, the subring generated by any subset of size at most two is a nilpotent Lie ring of nilpotency class at most three.
 * 2) The following identities hold for all $$x,y \in L$$:
 * 3) * $$[x,[x,[x,y]]] = 0$$
 * 4) * $$[x,[y,[x,y]]] = 0$$