Generalization of Killing form

Question

I am reading Knapp's book "Lie groups beyond introduction". On page 369, he has described the following. Let $\mathfrak g$ be a real semisimple Lie algebra. Suppose $\theta\colon\mathfrak g\to \mathfrak g$ is a Cartan involution. Let $B$ be a nondegenerate symmetric bilinear form on $\mathfrak g$ which is $\theta $-invariant and $B_\theta(X,Y):=-B(X,\theta Y)$ is positive definite. Let $\mathfrak g=\mathfrak k\oplus\mathfrak p$ be the Cartan decomposition of $\mathfrak g$ where $\mathfrak k$ and $\mathfrak p$ are eigenspaces of $\theta$ corresponding to eigenvalues $1$ and $-1$ respectively. Then clearly, $\mathfrak k\oplus i\mathfrak p$ is a compact real form of $\mathfrak g^{\mathbb C}$ (complexification of $\mathfrak g).$ Hence $B$ is negative definite on a maximal abelian subspace of $\mathfrak k\oplus i\mathfrak p$. I understood up to this point. Now Knapp argues that from above one can conclude that for any Cartan subalgebra of $\mathfrak g^{\mathbb C}$, $B$ is positive definite on the real subspace where all the roots are real valued. I do not understand how to get this. However, if $B$ is in particular the Cartan-Killing form then I can check by hand that this claim holds. Can someone please help me out?

Answer 1

According to theorem 2.15, all Cartan subalgebras of a complex semi-simple Lie algebras are conjugated. I.e. there exists $\alpha \in \mathrm{Inn}(\mathfrak{g})$ such that $\mathfrak{h}_1 = \alpha(\mathfrak{h}_2).$ Since any invariant form $B$ is also invariant with respect to the group of inner automorphisms (sorry, I don't know if or where Knapp proves this), it follows that any positivity property is preserved for corresponding real forms of all Cartan subalgebras.