Definition 2

Let G and H be Lie groups. A Lie group morphism of G into H (or simply a morphism of G into H if no confusion can arise) is a mapping of G into H which is a group homomorphism and is analytic. The automorphism group of G is denoted by $\operatorname{Aut}(G)$..

The identity mapping of G is a morphism. The composition of two morphisms is a morphism. If $f : G \to H$ and $f' : H \to G$ are two inverse morphisms, $f$ and $f'$ are Lie group isomorphisms.