Exercise 16, § 9

Let $G$ be a finite-dimensional Lie group. Show that there exists a basis $(x_1, \ldots, x_n)$ of $L(G)$ such that, for all $i$, $\exp(\mathbf{R}x_i)$ is a Lie subgroup of $G$. (Use Spectral Theories, Chapter II, § 2, Lemma 1.)