Direct sum of Lie algebras

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

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

[(X_1,Y_1),(X_2,Y_2)] := \bigl([X_1,X_2]_{\mathfrak g},\; [Y_1,Y_2]_{\mathfrak h}\bigr).

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

\mathrm{Lie}(G\times H)\cong \mathrm{Lie}(G)\oplus \mathrm{Lie}(H),

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.