# Restriction of the Killing form to a Cartan subalgebra is nondegenerate

Let $$\mathfrak g$$ be a semisimple Lie algebra over $$\mathbb{C}$$ with Cartan subalgebra $$\mathfrak h$$. Then the Killing form on $$\mathfrak g$$ defined by $$(X,Y) = \operatorname{Tr}(\operatorname{ad}(X) \circ \operatorname{ad}(Y))$$ is nondegenerate (this can also be taken as the definition of a semisimple Lie algebra).

I'm trying to understand the following:

1 . Why is the restriction of the Killing form to $$\mathfrak h$$ still nondegenerate?

2 . Why do we have $$(h,h) \neq 0$$ for all $$h \in \mathfrak h$$?

3 . Let $$\Phi \subseteq \mathfrak h^{\ast}$$ be the roots of $$\mathfrak h$$ in $$\mathfrak g$$. Let $$V$$ be the $$\mathbb{Q}$$-span of $$\Phi$$, and define a bilinear form on $$V$$ by $$(v,w) = (h_v,h_w)$$, where $$h_v \in \mathfrak h$$ is the inverse of $$v$$ under the $$\mathbb{C}$$-vector space isomorphism $$\mathfrak h \rightarrow \mathfrak h^{\ast}, h \mapsto (-,h)$$. Why does this define a symmetric, positive definite (hence nondegenerate) bilinear form on $$V$$?

The third statement is a claim made on page 1 of Steinberg, Lectures on Chevalley Groups.

Proof of 1: the linear transformations $$\operatorname{ad}(h) : h \in \mathfrak h$$ are simultaneously diagonalizable, with diagonal matrix $$\operatorname{diag}(0, \ldots , 0, \alpha(h) : \alpha \in \Phi)$$, where the number of zeros is the dimension of $$\mathfrak h$$. So the Killing form on $$\mathfrak h$$ is

$$(h,h') = \sum\limits_{\alpha \in \Phi} \alpha(h)\alpha(h') = 2\sum\limits_{\alpha \in \Phi^+} \alpha(h)\alpha(h')$$

where $$\Phi^+$$ is a given set of positive roots with base $$\Delta$$. Let $$h_1, \ldots , h_t$$ be a dual basis to simple roots $$\alpha_1, \ldots , \alpha_t$$. Fix $$1 \leq i \leq t$$, and each $$n \geq 1$$, let $$m_n$$ be the number of positive roots with the coefficient $$n\alpha_i$$. Then

$$(h,h_i) = 2(m_1 + 2m_2 + \cdots)\alpha_i(h)$$

So if $$(h,h') = 0$$ for all $$h' \in \mathfrak h$$, then $$\alpha_i(h) = 0$$ for all $$i$$ $$\Rightarrow h = 0$$.