Let $\mathfrak {g}$ be a finite dimensional complex simple Lie algebra.Show that
Let $\mathfrak {h}$ be a sub algebra of $\mathfrak {g}$.Show that $\mathfrak {h}$ is maximal toral sualgebra iff $C_{\mathfrak {g}}(\mathfrak {h})=\mathfrak {h}$
I know that if $\mathfrak {g}$ is semisimple then $C_{\mathfrak {g}}(\mathfrak {h})=\mathfrak {h}$ hence the forward direction of the above problem is clear.I am stuck in proving the other direction.Please help
If $L$ is a semisimple Lie algebra of characteristic zero, then maximal total subalgebras are precisely Cartan subalgebras, which are precisely abelian subalgebras $H$ with $N_L(H)=H$. For a proof, see Humphrey's book, page $80$ here. Abelian Cartan subalgebras are also self-centralising, see Theorem $8.2.1$ on page $29$ here.