Exercise 13, § 3

Let $\mathfrak{p}$ and $\mathfrak{q}$ be two parabolic subalgebras of a semi-simple Lie algebra $\mathfrak{a}$, and let $\mathfrak{n}$ be the nilpotent radical of $\mathfrak{p}$. Show that $\mathfrak{m}= (\mathfrak{p}\cap \mathfrak{q}) +\mathfrak{n}$ is a parabolic subalgebra of $\mathfrak{a}$. (Reduce to the case where $\mathfrak{a}$ is split and $\mathfrak{q}$ is a Borel subalgebra; choose a Cartan subalgebra $\mathfrak{h}$ contained in $\mathfrak{p}\cap \mathfrak{q}$, and determine the subset P of the corresponding root system such that $\mathfrak{m}=\mathfrak{h}+\mathfrak{g}^P.$)