Nondegeneracy of the Killing form on a Cartan subalgebra
= Nondegeneracy of the Killing form on a Cartan subalgebra
For a complex <semisimple Lie algebra>, the <Cartan subalgebra> is orthogonal under the <Killing form> to every nonzero <root space>. A vector in the Cartan subalgebra orthogonal to that whole subalgebra is therefore orthogonal to the entire Lie algebra and must vanish. On the Cartan subalgebra the form is $\kappa(h,h')=\sum_{\alpha\in\Phi}\alpha(h)\alpha(h')$.