CS-Baer 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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Definition

A CS-Baer Lie ring is a Lie ring $L$ satisfying the following two conditions:

• $L$ is a Lie ring of nilpotency class two, i.e., the derived subring $[L,L]$ is contained in the center $Z(L)$.
• There is a Lie subring $K$ of $L$ such that $[L,L]\leq K\leq Z(L)$ and such that every element of $[L,L]$ has a unique half in $K$. Note that since $Z(L)$ is an abelian Lie ring, any additive subgroup of $Z(L)$ gives a Lie subring contained in $Z(L)$.

Relation with other properties

Stronger properties

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

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Lie ring whose bracket is the double of a Lie bracket giving nilpotency class two				|FULL LIST, MORE INFO