Exercise 6, § 3

Two Borel subalgebras of a semi-simple Lie algebra $\mathfrak{a}$ are said to be opposite if their intersection is a Cartan subalgebra. Show that, if $\mathfrak{b}$ is a Borel subalgebra of $\mathfrak{a}$, and $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{b}$, there exists a unique Borel subalgebra of $\mathfrak{a}$ that is opposite to $\mathfrak{b}$ and contains $\mathfrak{h}$. (Reduce to the split case.)