Nilpotency class three is 3-local for Lie rings

Statement
The following are equivalent for a Lie ring $$L$$:


 * 1) The nilpotency class of $$L$$ is at most three, i.e., we have $$[w,[x,[y,z]]] = 0$$ for all $$w,x,y,z \in L$$ (where we allow some or all of $$w,x,y,z$$ to be equal).
 * 2) The 3-local nilpotency class of $$L$$ is at most three, i.e., the subring generated by any three elements of $$L$$ is nilpotent of class at most three.

Similar facts

 * 2-Engel implies class three for Lie rings

Applications

 * n-local nilpotency class n implies nilpotency class n

Facts used

 * 1) uses::Polarization trick: In particular, this states that alternating implies skew-symmetric.

(1) implies (2)
This is immediate.

(2) implies (1)
The proof is incomplete.

Given: A Lie ring $$L$$ of 3-local nilpotency class three

To prove: $$[w,[x,[y,z]]] = 0$$ for all $$w,x,y,z \in L$$

Proof: