This article defines a Lie ring property: a property that can be evaluated to true/false for any Lie ring.
View a complete list of properties of Lie rings
VIEW RELATED: Lie ring property implications  Lie ring property nonimplications Lie ring metaproperty satisfactions  Lie ring metaproperty dissatisfactions  Lie ring property satisfactions  Lie ring property dissatisfactions
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: 2Engel group
View other analogues of 2Engel group  View other analogues in Lie rings of group properties (OR, View as a tabulated list)
Definition
A 2Engel Lie ring can be defined in the following equivalent ways:
No. 
Shorthand 
A Lie ring is termed a 2Engel Lie ring if ...

1 
2Engel identity 
for any , we have (Note that if , this would follow automatically, so we can restrict attention to the case ).

2 
triple Lie bracket is alternating 
The function is an alternating function in all pairs of variables.

3 
2locally class at most two 
any subring of 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.

4 
cyclic symmetry of Lie bracket 
for any , we have . Note that are possibly equal, possibly distinct.

5 
union of abelian ideals 
There is a collection (for some indexing set ) of abelian ideals of such that . Equivalently, every element of is contained in an abelian ideal.

Relation with other properties
Stronger properties
Weaker properties
Facts