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

Statement
Suppose $$L$$ is a Lie ring that satisfies the following two conditions:


 * $$L$$ is a uses property satisfaction of::2-Engel Lie ring, i.e., $$[x,[x,y]] = 0$$ for all $$x,y \in L$$.
 * $$3x = 0$$ implies $$x = 0$$, i.e., $$L$$ is free of 3-torsion.

Then, $$L$$ is a proves property satisfaction of::Lie ring of nilpotency class two, i.e., $$[x,[y,z]] = 0$$ for all $$x,y,z \in L$$.

Similar facts for 2-Engel conditions

 * 2-Engel and 3-torsion-free implies class two for groups
 * 2-Engel and Lazard Lie ring implies class two
 * 2-Engel and Lazard Lie group implies class two
 * 2-Engel implies class three for Lie rings
 * 2-Engel implies class three for groups
 * Nilpotency class three is 3-local for Lie rings

Similar facts for higher Engel conditions

 * 3-Engel and 2-torsion-free implies 2-local class three for Lie rings
 * 3-Engel and (2,5)-torsion-free implies class six for Lie rings

Facts used

 * 1) uses::2-Engel implies third member of lower central series is in 3-torsion for Lie rings

Proof
Given: A 2-Engel Lie ring $$L$$ that is 3-torsion-free.

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

Proof: By Fact (1), we have that $$3[x,[y,z]] = 0$$ for all $$x,y,z \in L$$. The result now follows immediately from the 3-torsion-free assumption.