Class two Lie cring

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Definition

A class two Lie cring is an abelian group $L$ (denoted additively) equipped with an additional binary operation $*$ satisfying the following additional conditions:

Condition name	Description
$*$ is a 2-cocycle for trivial group action from $L$ to itself	$(a*(b+c))+(b*c)=((a+b)*c)+(a*b)$ for all $a,b,c\in L$
$*$ is cyclicity-preserving	$a*b=0$ if $⟨a,b⟩$ is cyclic.
$*$ is skew-symmetric	$a*b=-(b*a)$ for all $a,b\in L$
$*$ has class two	$a*(b*c)=0$ for all $a,b,c\in L$