Skew of class two near-Lie cring is class two Lie ring

Statement

Suppose $L$ is a Class two near-Lie cring (?) with abelian group structure denoted additively and cring operation denoted multiplicatively. Then, with the same additive group, and with Lie bracket:

$\![x,y]:=(x*y)-(y*x)$

$L$ acquires the structure of a Lie ring of nilpotency class two.

Facts used

1. Skew of 2-cocycle for trivial group action of abelian group is alternating bihomomorphism

Related facts

• Double of class two Lie cring is class two Lie ring: In the special case that the cring structure is skew-symmetric (so we get a class two Lie cring) the skew becomes the double map.