Internal semidirect product of Lie rings

Definition
Suppose $$L$$ is a Lie ring and $$A$$ and $$B$$ are Lie subrings of $$L$$. We say that $$L$$ is an internal semidirect product of $$A$$ and $$B$$, denoted $$L = A \rtimes B$$, if it satisfies all the following conditions:


 * 1) $$A$$ is an ideal in $$L$$.
 * 2) $$B$$ is a subring in $$L$$.
 * 3) $$A \cap B$$ is the zero subring of $$L$$, i.e., it is the set $$\{ 0 \}$$.
 * 4) $$L = A + B$$.

Conditions (3) and (4) basically say that the additive group of $$L$$ is the internal direct product of the additive groups of $$A$$ and $$B$$.

This notion is equivalent to the notion of external semidirect product of Lie rings, via equivalence of internal and external semidirect product for Lie rings.