I could not get the following, could someone give me a hint?
Let $\mathfrak{H}$ be a Cartan subalgebra of a simple Lie algebra $\mathfrak{L}$. Show that $\mathfrak{H}$ is abelian.
So, we need to prove that $[\mathfrak{H},\mathfrak{H}]=0$. It seems that I should find a proper ideal of $\mathfrak{L}$ containing $[\mathfrak{H},\mathfrak{H}]$ but then I can not get the way.
Thanks in advance.