The displayed basis obeys $[\tau_a,\tau_b]=\epsilon_{abc}\tau_c$. Therefore
$$
\kappa(\tau_a,\tau_b)
=\epsilon_{ac d}\epsilon_{bd c}=-2\delta_{ab}.
$$
The basis is adapted to the Killing form in the usual orthogonal-basis sense, since its Gram matrix is diagonal; rescaling by $1/\sqrt2$ makes it orthonormal for $-\kappa$.


Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-302/2/d/solution