Exercise 5, § 11

Assume that $\mathfrak g$ is semi-simple. Let $\mathfrak a$ be a subalgebra of $\mathfrak g$, reductive in $\mathfrak g$, of the same rank as $\mathfrak g$, and containing a principal $\mathfrak{sl}_2$-triplet of $\mathfrak g$. Show that $\mathfrak a=\mathfrak g$.