2-Engel and Lazard Lie ring implies class two

Statement
Suppose $$L$$ is a Lie ring satisfying the following two conditions: it is both a fact about::2-Engel Lie ring and a fact about::Lazard Lie ring. Then, $$L$$ must be a fact about::Lie ring of nilpotency class two.

Related facts

 * 2-Engel and Lazard Lie group implies class two
 * 2-Engel and 3-torsion-free implies class two