Definition
Category of real Lie groups
Finite-dimensional real Lie groups and smooth group homomorphisms.
The category of real Lie groups has finite-dimensional smooth real Lie groups as objects and smooth group homomorphisms as morphisms. Manifolds are Hausdorff and second countable. The identity maps and compositions are smooth group homomorphisms.
Underlying real groups
A complex Lie group defines an object here by forgetting its complex structure. Smooth real Lie-group maps need not be holomorphic. Entrywise conjugation on illustrates the difference.
Linear terminology
A matrix Lie group is generally not a vector subspace: it is not closed under addition or scalar multiplication. Thus “real-linear endomorphism of a Lie group” does not name the morphisms of this category. Use smooth group maps here, or specify an underlying Lie algebra, a representation space, or an ambient matrix-space extension when linearity is intended.