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

Statement
Suppose $$L$$ is a fact about::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) uses::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.