Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 3 / 3 / iv / Solution

For a finite-dimensional Lie algebra, the Killing form is the symmetric bilinear formIn the root-space decomposition of this semisimple Lie algebra, $\mathrm{ad}H$ acts as zero on the abelian Cartan subalgebra and as multiplication by $\alpha (H)$ on ${\mathfrak{g}}_{\alpha }$. For $H,K\in \mathfrak{h}$, their product thus acts as multiplication by $\alpha (H)\alpha (K)$ on each root space. Every root space here is one-dimensional. Taking the trace over the two-dimensional Cartan subalgebra and the eight root spaces provesFor a general weight decomposition the trace would include the corresponding weight multiplicities; the root multiplicities are one in the semisimple setting here.