Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 2 / 5 / ii

Use the standard coroot normalization $\alpha (H_{\alpha })=2$. The symbols $H_{\alpha }$ are vectors in the Cartan subalgebra, not the root-space vectors $X_{\alpha }$. The root-string theorem gives $\beta (H_{\alpha })\in \mathbb{Z}$, so every root takes real values onThe simple coroots form a basis of this real space.

In the root-space decomposition, $\mathrm{ad}H$ is zero on $\mathfrak{h}$ and scalar multiplication by $\beta (H)$ on the one-dimensional space ${\mathfrak{g}}_{\beta }$. Therefore, for $H,K\in \mathfrak{h}$,For $H,K\in {\mathfrak{h}}_{\mathbb{R}}$ every term is real, so the restricted form is real. Moreover,If it is zero, all $\beta (H)$ vanish. Then $H$ commutes with every root space and with $\mathfrak{h}$, hence is central in $\mathfrak{g}$. The center of a Lie algebra vanishes for a semisimple Lie algebra, so $H=0$. Thus the Euclidean subspace of a Cartan subalgebra hasEquivalently, positivity follows because the roots span ${\mathfrak{h}}^{\ast }$.