Exercise 5, § 3

Let $\mathfrak{m}$ be a parabolic subalgebra of a semi-simple Lie algebra $\mathfrak{a}$.

a) Let $\mathfrak{p}$ be a subalgebra of $\mathfrak{m}$. Then $\mathfrak{p}$ is a parabolic subalgebra of $\mathfrak{a}$ if and only if $\mathfrak{p}$ contains the radical $\mathfrak{r}$ of $\mathfrak{m}$ and $\mathfrak{p}/\mathfrak{r}$ is a parabolic subalgebra of the semi-simple algebra $\mathfrak{m}/\mathfrak{r}$.

b) If $\mathfrak{m}'$ is a parabolic subalgebra of $\mathfrak{a}$, every Cartan subalgebra of $\mathfrak{m}\cap \mathfrak{m}'$ is a Cartan subalgebra of $\mathfrak{a}$. (Reduce to the split case, and apply Prop. 10 to the Borel subalgebras contained in $\mathfrak{m}$ and $\mathfrak{m}'.$)