Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 302 / 3 / c / Solution

Because the translations commute, $P^{\sigma }{P}^{\mu }$ is symmetric in $(\sigma ,\mu )$, whereas the Levi-Civita tensor in the Pauli-Lubanski pseudovector is antisymmetric, so $W_{\mu }{P}^{\mu }=0$. Moreover,because the two terms cancel after relabeling and each remaining momentum product is symmetric.