2-Engel Lie ring

This article defines a Lie ring property: a property that can be evaluated to true/false for any Lie ring.
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ANALOGY: This is an analogue in Lie ring of a property encountered in group. Specifically, it is a Lie ring property analogous to the group property: 2-Engel group
View other analogues of 2-Engel group | View other analogues in Lie rings of group properties (OR, View as a tabulated list)

Definition

A 2-Engel Lie ring can be defined in the following equivalent ways:

No.	Shorthand	A Lie ring $L$ is termed a 2-Engel Lie ring if ...
1	2-locally class at most two	any subring of $L$ generated by a subset of size at most two is a Lie ring of nilpotency class two, i.e., any such subring has class at most two.
2	2-Engel identity	for any $x,y\in L$, we have $[x,[x,y]]=0$ (Note that if $x=y$, this would follow automatically, so we can restrict attention to the case $x\ne y$).

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
abelian Lie ring	Lie bracket of any two elements is trivial			|FULL LIST, MORE INFO
Lie ring of nilpotency class two	$[x,[y,z]]=0$ for any (not necessarily distinct) elements $x,y,z$ of the Lie ring			|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Engel Lie ring				|FULL LIST, MORE INFO

Facts

• Engel's theorem
• Kostrikin's theorem
• 2-Engel and Lazard Lie ring implies class two
• 2-Engel and 3-torsion-free implies class two