Exercise 29, § 6

Let G be a finite-dimensional complex connected Lie group, G_0 the underlying real Lie group and H an integral subgroup of G_0. Show that there exists a smallest integral subgroup H^* of G containing H. Give an example where H is closed in G_0 but H^* is not closed in G (take G = \mathbf{C}^2/\mathbf{Z}^2).