Lie functor commutes with realification

Statement

For every complex Lie group $G$ , there is a natural isomorphism of real Lie algebras

\operatorname{Lie}_{\mathbb R}(G_{\mathbb R}) \cong \bigl(\operatorname{Lie}_{\mathbb C}G\bigr)_{\mathbb R}.

Here $G_{\mathbb R}$ is the underlying real Lie group and the expression on the right is the underlying real Lie algebra.

Why the identification holds

Both sides have the same real tangent space $T_eG$ . Their brackets are obtained from the same left-invariant vector fields and therefore agree. Naturality means that for a holomorphic homomorphism $f:G\to H$ , this identification intertwines the real differential of $f_{\mathbb R}$ with the scalar restriction of the complex differential of $f$ .

Consequently, if $G$ has complex dimension $n$ , its complex Lie algebra has complex dimension $n$ , while the Lie algebra of $G_{\mathbb R}$ has real dimension $2n$ .

References

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Birkhäuser, 2002, §I.3. Publisher record.
Brian C. Hall, Lie Groups, Lie Algebras, and Representations, 2nd ed., Springer, 2015, Chapters 2–3. Publisher record.