# Internal semidirect product of Lie rings

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 $B$.