Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 302 / 2 / d / Solution

The displayed basis obeys $[{\tau }_a,{\tau }_b]={ϵ}_{abc}{\tau }_c$. ThereforeThe basis is adapted to the Killing form in the usual orthogonal-basis sense, since its Gram matrix is diagonal; rescaling by $1/\sqrt{2}$ makes it orthonormal for $-\kappa$.