0
$\begingroup$

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.

  • 2
    The definition of Cartan subalgebra is not standard. What is your definition?2012-11-02
  • 0
    Definition : $\mathfrak{H}$ is a Cartan subalgebra if $\mathfrak{H}$ is nilpotent and self-normalizing, i.e., if $[x,y]\in \mathfrak{H}$ for all $x\in\mathfrak{H}$ then $y\in\mathfrak{H}$.2012-11-10

1 Answers 1