Given Lie groups G and H (see Lie group), their product Lie group is the manifold G×H with group structure
(g,h)⋅(g′,h′)=(gg′,hh′),(g,h)−1=(g−1,h−1).
With the product smooth structure, the multiplication and inversion maps are smooth, so G×H is a Lie group. The coordinate projections
prG:G×H→G,prH:G×H→H
are smooth group homomorphisms (see Lie group homomorphism).
On the infinitesimal level, the Lie algebra satisfies
Lie(G×H)≅Lie(G)⊕Lie(H),
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:
expG×H(X,Y)=(expG(X),expH(Y)).
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).