Lie algebra of real algebraic group equals Lie algebra of corresponding real Lie group

Hands-on statement
Suppose $$G$$ is a real algebraic group, i.e., an algebraic group over the field of real numbers. Denote by $$\mathfrak{g}$$ the corresponding Lie algebra in the algebraic group sense. Then, $$\mathfrak{g}$$ can be naturally identified with the Lie algebra of $$G$$ viewed as a real Lie group.

Category theory statement
We have a commuting triangle, i.e., the following is true in a canonical sense:

(Lie algebra functor for real Lie groups) $$\circ$$ (Forgetful functor from real algebraic groups to real Lie groups) = (Lie algebra functor for algebraic groups over the reals)