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 $\kappa (h,h^{\prime })={\sum }_{\alpha \in \Phi }\alpha (h)\alpha (h^{\prime })$.

The real span of the simple coroots inside a complex Cartan subalgebra carries a positive-definite restriction of the Killing form. Equivalently it consists of the $h$ for which all roots have real values. The formula $\kappa (h,h)={\sum }_{\alpha \in \Phi }\alpha (h{)}^2$ is positive for $h\ne 0$, since the roots span the dual. The induced dual inner product makes the root system Euclidean and makes root reflections orthogonal.

Let $\mathcal{B}(H)=B(H,\cdot )$ be the Killing form identification of a Cartan subalgebra with its dual. The root-space decomposition gives $B(H,K)={\sum }_{\beta \in \Phi }\beta (H)\beta (K)$, and therefore the displayed root expansion. On the real coroot span its coefficients are real, proving $\mathcal{B}({\mathfrak{h}}_{\mathbb{R}})\subseteq \mathbb{R}P$. For ${\mathfrak{sl}}_3$ and $H_{12}=E_{11}-E_{22}$, the formula gives $\mathcal{B}(H_{12})=6(L_1-L_2)$. The normalization is $B(X,Y)=\mathrm{tr}(\mathrm{ad}X\mathrm{ad}Y)$, not a rescaled trace form.

// Target: semisimple-lie-algebra.bigb