Lie ring arising as the skew of a class two near-Lie cring

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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

Definition

A Lie ring arising as the skew of a class two near-Lie cring is defined as a Lie ring L such that there exists a class two near-Lie cring structure with the same underlying set and additive group structure, such that, if * denotes the cring operation, then:

\! [x,y] = (x * y) - (y * x) \ \forall \ x,y \in L

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
abelian Lie ring
Baer Lie ring
LCS-Baer Lie ring
Lie ring whose bracket is the double of a Lie bracket giving nilpotency class two
Lie ring arising as the double of a class two Lie cring

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Lie ring of nilpotency class two double of class two Lie cring is class two Lie ring