Semidirect product of Lie algebras

The semidirect product of Lie algebras is a generalization of the Direct product of Lie algebras where only one of the operands is required to be an ideal.¹. The semidirect product is an extension of by ,

and extensions which can be written this way are precisely split extensions.

Internal semidirect product.

Let and be subalgebras, the first of which is an ideal, such that internally. Then is the internal semidirect product .

Let be a Lie algebra and let be a Lie algebra acting on by derivations, i.e. equipped with a Lie algebra homomorphism into the Derivation subalgebra , so that is a derivation of for every . Then the external semidirect product is the unique Lie bracket on the sum vector space such that and are subalgebras and lie

for all and .