I am studying Theorem 10 on page 147 of Jaboson's Lie algebra book, and at the end of the proof I believe he uses the fact that the restriccion of the Killing form to $H$, the Cartan subalgebra is positive definite, so my question is:
If L is a semisimple Lie algebra over an algebraic closed field $F$ of characteristic 0 and $H$ is a Cartan subalgebra, is the Killing form positive definite on $H$?
If so, how can I prove that? (or where can I find the proof?) I know how to prove that it is non degenerate.
If not, why does Jacobson assume so, or under what hypothesis is that so?
I did the example $L=sl(2,F)$ and it is definite positive on $H$.
The statement in Theorem $10$ to prove is that the real Lie algebra $L_u$ is compact. So it has to be shown that its Killing form over the real numbers is negative definite. Jacobson defines just before Theorem $10$ a real Lie algebra to be compact if its Killing form is negative definite. This implies semisimple.