Direct sum of Lie algebras

Let $\mathfrak g$ and $\mathfrak h$ be Lie algebras over the same field.

Definition

The direct sum Lie algebra $\mathfrak g\oplus \mathfrak h$ is the direct sum as vector spaces equipped with the bracket

With this bracket, the inclusions $\mathfrak g\hookrightarrow \mathfrak g\oplus\mathfrak h$ and $\mathfrak h\hookrightarrow \mathfrak g\oplus\mathfrak h$ are Lie algebra homomorphisms, and $\mathfrak g$ and $\mathfrak h$ commute inside the sum.

Universal property

Giving a Lie algebra homomorphism $\mathfrak g\oplus\mathfrak h\to\mathfrak k$ is equivalent to giving homomorphisms $\mathfrak g\to\mathfrak k$ and $\mathfrak h\to\mathfrak k$ whose images commute.

Relation to Lie groups

If $G$ and $H$ are Lie groups, then the Lie algebra of the product Lie group $G\times H$ is canonically

as recorded in Lie algebra of a product .

Context. Many decomposition results (e.g. semisimple as a direct sum of simples ) are literally statements that a Lie algebra splits as a direct sum of ideals.