Lie bracket of two subrings

Definition
Suppose $$L$$ is a Lie ring and $$A,B$$ are subrings of $$L$$. The Lie bracket of $$A$$ and $$B$$, denoted $$[A,B]$$ is the additive subgroup of $$L$$ generated by all elements of the form $$[a,b]$$ where $$a \in A, b \in B$$.

Note that the Lie bracket of subrings need not be a subring. If both the subrings are ideals, the Lie bracket is again an ideal. If, further, both the subrings are derivation-invariant subrings, the Lie bracket is also derivation-invariant.