Lie ring of nilpotency class three

Definition
A Lie ring of nilpotency class three is a Lie ring $$L$$ satisfying the following equivalent conditions:


 * 1) Its nilpotency class is at most three. This is equivalent to checking the identity: $$\! [w,[x,[y,z]]] = 0 \ \forall \ w,x,y,z \in L$$
 * 2) Its 3-local nilpotency class is at most three. In other words, the subring generated by any subset of size at most three is a nilpotent Lie ring of nilpotency class at most three.
 * 3) The following identities hold for all $$x,y,z \in L$$:
 * 4) * $$[x,[y,[y,z]]] = 0$$
 * 5) * $$[x,[y,[x,z]]] = 0$$
 * 6) The Lie ring is a 2-bi-Engel Lie ring, i.e., the following hold for all $$u,x,y \in L$$:
 * 7) * $$u,[u,x,y] = 0$$
 * 8) * $$u,x],[u,y = 0$$
 * 9) * $$[x,[u,[u,y]]] = 0$$