Exercise 3, § 3

Let $\mathfrak{g}$ be a real Lie algebra, $\mathfrak{r}$ the radical of $\mathfrak{g},\mathfrak{h}$ and $\mathfrak{h}'$ Cartan subalgebras of $\mathfrak{g},\varphi$ the canonical homomorphism from $\mathfrak{g}$ to $\mathfrak{g}/\mathfrak{r}$. The following conditions are equivalent:

(i) $\mathfrak{h}$ and $\mathfrak{h}'$ are conjugate under Int($\mathfrak{g}$);

(ii) $\varphi (\mathfrak{h})$ and $\varphi (\mathfrak{h}')$ are conjugate under Int($\mathfrak{g}/\mathfrak{r}$). (Imitate the proof of Prop. 5.)