Exercise 7, § 6

Let $\phi$ be the morphism of $\mathbf{Z}/2\mathbf{Z}$ into the permutation group of $\mathbf{C}$ which maps the non-identity element to $z \mapsto \bar{z}$. Let $G$ be the semi-direct product of $\mathbf{Z}/2\mathbf{Z}$ by the real Lie group $\mathbf{C}$ corresponding to $\phi$. It is a real Lie group with lie algebra $\mathbf{R}^2$. Show that there exists on $G$ no complex Lie group structure compatible with the real Lie group structure.