Exercise 21, § 6

Let $G$ be a real Lie group and $H$ a normal 1-dimensional integral subgroup. (a) If $H$ is not closed, $\overline{H}$ is compact (Spectral Theories, Chapter II, § 2, Lemma 1), hence isomorphic to $T^n$ and hence central in $G$ if $G$ is connected (use General Topology, Chapter VII, § 2, Proposition 5). (b) Suppose that $H$ is closed. Let $a$ be an element of $G$ and $C(a)$ the set of elements of $G$ which commute with $a$. Suppose that $H \not\subset C(a)$. If $H$ is isomorphic to $\mathbf{R}$, then $C(a) \cap H = \{e\}$. (Consider the automorphism $\alpha : h \mapsto a^{-1}ha$ of $H$.) If $H$ is isomorphic to $\mathbf{T}$, $C(a) \cap H$ has two elements. (Consider $\alpha$ again and use General Topology, Chapter VII, § 2, Proposition 6.) The second situation is impossible if $G$ is connected (use (a)). (c) With the hypotheses of (b), suppose further that $G/H$ is commutative.

Then $G = C(a).H.$ (Observe that the mapping $h \mapsto h^{-1}\alpha(h)$ of $H$ into $H$ is surjective.)