Proposition 7

Let V be a neighbourhood of $e$ in G. Every Cartan subalgebra of $\mathfrak{g}$ is of the form $\mathfrak{g}^1(a)$ where $a$ is a regular element of G belonging to V.

By Prop. 6, there exists an open neighbourhood U of 0 in $\mathfrak{g}$ and an exponential map exp$: U\rightarrow G$ satisfying the conditions of Lemma 4. If $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$, there exists a regular element $x\in \mathfrak{h}$ such that $\mathfrak{h}=\mathfrak{g}^0(x) ($§3, Th. 2). On the other hand, there exists an element $t\in k^*$ such that $tx\in U$ and exp($tx$)$\in V$. Then $\mathfrak{h}=\mathfrak{g}^0(x) =\mathfrak{g}^0(tx) =\mathfrak{g}^1$(exp($tx$)), and by Lemma 4 (ii), exp($tx$) is a regular element of G.