Zelmanov's theorem on Engel Lie rings

Statement
The statement has many equivalent versions:


 * 1) For any natural number $$n$$, any finitely generated $$n$$-Engel Lie ring is a nilpotent Lie ring. Note that finite generation is as a Lie ring, not necessarily as an abelian group.
 * 2) For any natural number $$n$$, any $$n$$-Engel Lie ring is a locally nilpotent Lie ring.
 * 3) A Lie ring is a locally nilpotent Lie ring if and only it is a (not necessarily bounded) Engel Lie ring.

Similar facts for Lie rings

 * Engel's theorem is a similar result for finite-dimensional Lie algebras over fields
 * 2-Engel implies class three for Lie rings
 * 2-Engel and 3-torsion-free implies class two for Lie rings
 * 3-Engel and (2,5)-torsion-free implies class six for Lie rings

Similar facts for groups

 * Zorn's theorem says that finite Engel groups are nilpotent
 * 2-Engel implies class three for groups
 * 2-Engel and 3-torsion-free implies class two for groups
 * 3-Engel and (2,5)-torsion-free implies class four for groups
 * 3-Engel implies locally nilpotent for groups
 * 4-Engel and (2,3,5)-torsion-free implies class seven for groups
 * 4-Engel implies locally nilpotent for groups
 * The questions are open for $$n$$-Engel groups, $$n \ge 5$$.