Exercise 19, § 9

Let G be a locally compact group and H a closed subgroup of G. Suppose that H is a finite-dimensional simply connected solvable Lie group and that $G/H$ is compact. Show that there exists a compact subgroup L of G such that G is the semi-direct product of L and H. (Argue by induction on dim H. Consider the last non-trivial derived group of H and use Integration, Chapter VII, § 3, Proposition 3.)