2-cocycle for trivial Lie ring action

Definition
Suppose $$L$$ is a Lie ring and $$M$$ is an abelian group.

Explicit definition
A 2-cocycle for trivial Lie ring action is a homomorphism of groups $$f: L \times L \to M$$ (where $$L \times L$$ is the external direct product of $$L$$ with itself) satisfying the following additional condition:

$$\! f(x,[y,z]) = f([x,y],z) + f(y,[x,z]) \ \forall \ x,y,z \in L$$

Definition in terms of 2-cocycle
A 2-coycle for trivial Lie ring action is a special case of a defining ingredient::2-cocycle for a Lie ring action in the case where the action is trivial.