A Lie subgroup of a Lie group is usually defined as a subgroup which is also a submanifold. But actually any closed subgroup of a Lie group is automatically a manifold, hence a Lie subgroup. Similarly, any continuous group homomorphism between Lie groups is automatically smooth, i.e. a morphism in the category of Lie groups.

A Lie subgroup of a Lie group is usually defined as a subgroup which is also a submanifold. But actually any closed subgroup of a Lie group is automatically a manifold, hence a Lie subgroup. Similarly, any continuous group homomorphism between Lie groups is automatically smooth, i.e. a morphism in the category of Lie groups.

NB Needless to say, the usual definitions (maybe due to Chevalley?) are there for very good reasons and shouldn't be changed. I just find it interesting that formally weakening them doesn't change the concepts studied.