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 non-implications |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: 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)
A 2-Engel Lie ring can be defined in the following equivalent ways:
|| A Lie ring is termed a 2-Engel Lie ring if ...
|| 2-Engel identity
|| for any , we have (Note that if , this would follow automatically, so we can restrict attention to the case ).
|| triple Lie bracket is alternating
|| The function is an alternating function in all pairs of variables.
|| 2-locally 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.
|| cyclic symmetry of Lie bracket
|| for any , we have . Note that are possibly equal, possibly distinct.
|| 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