Definition
Category of Lie algebras over a field
Lie algebras over a fixed field and bracket-preserving linear maps.
Fix a field . The category has Lie algebras over as objects and -linear maps satisfying
as morphisms. Identities and composition preserve linearity and the bracket.
Scalars and invertibility
A complex Lie algebra has both a complex-linear and an underlying real-linear Lie-algebra view. Forgetting the bracket gives still larger linear Hom-spaces. In either Lie-algebra category, automorphisms are the invertible bracket-preserving maps. The zero map is an endomorphism, but is an automorphism only for the zero Lie algebra.
Groups are different objects
Differentiation sends a Lie-group homomorphism to a Lie-algebra homomorphism. Integrating a Lie-algebra map requires global information about the groups; equal Lie algebras do not imply isomorphic Lie groups.