2-Engel Lie ring implies third member of lower central series is in 3-torsion

Statement
Suppose $$L$$ is a Lie ring that is a uses property satisfaction of::2-Engel Lie ring, i.e., $$[x,[x,y]] = 0$$ for all $$x,y \in L$$. Note that, by equivalence of definitions of 2-Engel Lie ring, this is equivalent to saying that $$[x,[y,z]] = [y,[z,x]] = [z,[x,y]]$$ for all $$x,y,z \in L$$.

Then, the third member of the lower central series of $$L$$, i.e., $$[L,[L,L]]$$, is in the 3-torsion of $$L$$. In other words, $$3[L,[L,L]] = 0$$.

Applications

 * 2-Engel and 3-torsion-free implies class two for Lie rings

Proof
Given: A Lie ring $$L$$ such that $$[x,[y,z]] = [y,[z,x]] = [z,[x,y]]$$ for all $$x,y,z \in L$$.

To prove: $$3[x,[y,z]] = 0$$ for all $$x,y,z \in L$$.

Proof: The proof follows by plugging the given data into Jacobi's identity, which states that:

$$[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0$$ for all $$x,y,z \in L$$