Product Lie group

Given Lie groups $G$ and $H$ (see Lie group), their product Lie group is the manifold $G\times H$ with group structure

With the product smooth structure, the multiplication and inversion maps are smooth, so $G\times H$ is a Lie group. The coordinate projections

are smooth group homomorphisms (see Lie group homomorphism).

On the infinitesimal level, the Lie algebra satisfies

compatibly with brackets (see Lie algebra of a product and direct sum of Lie algebras). Under this identification, the exponential map (see exponential map) splits:

This construction is ubiquitous: representation theory often reduces statements about a product to separate statements about factors (compare with tensor products of representations and irreducibles).