# 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? – Vít Tuček 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. – A beginner mathmatician Jun 15 '20 at 13:34
• No, it's not all right. What do you mean by "all roots are real values"? – Vít Tuček Jun 15 '20 at 13:45
• @Vit. Corrected. – A beginner mathmatician 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.