Lie cring

Definition

A Lie cring is an abelian group $L$ (denoted additively) equipped with a 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 $\langle a,b\rangle$ is cyclic.
$*$ is skew-symmetric	$a*b=-(b*a)$ for all $a,b\in L$
$*$ satisfies the Jacobi identities	$(a*(b*c))+(b*(c*a))+(c*(a*b))=0$ and $((a*b)*c)+((b*c)*a)=0$ for all $a,b,c\in L$

Equivalently, a Lie cring is a near-Lie cring for which the cring operation $*$ is skew-symmetric.