External semidirect product of Lie rings

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.