External semidirect product of Lie rings

Definition

Definition with the left action convention

Suppose $N$ is a Lie ring and $H$ is a Lie ring along with a homomorphism of Lie rings $\varphi$ from $H$ to the Lie ring of derivations $\operatorname {Der} (N)$. The external semidirect product of Lie rings, denoted $N\rtimes H$, is defined as follows:

• As an abelian group, it is the external direct product of $N$ and $H$, denoted $N\times H$ or $N\oplus H$.
• The Lie bracket is as follows:

$[(n_{1},h_{1}),(n_{1},h_{2})]=[n_{1}+\varphi (h_{1})n_{2},h_{1}+h_{2}]$

where $\varphi (h_{1})n_{2}$ denotes the image of $n_{2}$ under the derivation $\varphi (h_{1})$.

Why the action convention does not matter

This is basically because every Lie ring is naturally isomorphic to its opposite Lie ring via the negative map.