Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 304 / 4 / a / Solution

Treat the BRST transformation $Q$ as an odd graded derivation. In matrix notation the first three transformations areThenafter substituting $Qc$ and using the Jacobi identity. Similarly, the two terms incancel because the ghost components anticommute and only the Lie-algebra commutator survives. Applying $Q$ once more to $Qc$ gives a sum proportional to $f^d{}_{e[a}{f}^e{}_{bc]}{c}^a{c}^b{c}^c$, which vanishes by Jacobi. Finally,Thus $Q^2=0$ on every field.