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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?

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    $\begingroup$ 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? $\endgroup$ – Vít Tuček Jun 15 '20 at 13:26
  • $\begingroup$ @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. $\endgroup$ – A beginner mathmatician Jun 15 '20 at 13:34
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    $\begingroup$ No, it's not all right. What do you mean by "all roots are real values"? $\endgroup$ – Vít Tuček Jun 15 '20 at 13:45
  • $\begingroup$ @Vit. Corrected. $\endgroup$ – A beginner mathmatician Jun 15 '20 at 15:03
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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.

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