# Generalization of Killing form

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?

• I don't understand your question. Please check for typos. I've consulted Knapp's book and page 369 in my edition (from 1996) is devoted to exercies. Which edition are you using? Jun 15 '20 at 13:26
• @Vit. I am reading the second edition. Please help me out if you can. I am struggling to understand this point. As far as typos are concerned, I think that is alright. Jun 15 '20 at 13:34
• No, it's not all right. What do you mean by "all roots are real values"? Jun 15 '20 at 13:45
• @Vit. Corrected. Jun 15 '20 at 15:03

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.