External semidirect product of Lie rings

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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:

[(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.