Lie ring arising as the double of a class two 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 double of a class two Lie cring is defined as a Lie ring L such that there exists the structure of a class two Lie cring with the same additive group and underlying set L, such that the Lie bracket of L is the double of the cring operation, i.e., if the cring operation is *, then:

[x,y] = 2(x * y) \ \forall \ x,y \in L

Relation with other properties

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

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Lie ring arising as the skew of a class two near-Lie cring
Lie ring of nilpotency class two double of class two Lie cring is class two Lie ring