Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 305 / 1 / a / Solution

The first operator is a Lorentz scalar formed from commuting translations. Since $[P_{\mu },P_{\nu }]=0$,and the vector transformation law of $P^{\mu }$ gives

The Pauli-Lubanski pseudovector commutes with every $P_{\mu }$, so $[W^2,P_{\mu }]=0$. Its Lorentz-vector commutator similarly givesbecause the two tensor-rotation terms cancel after relabelling the contracted index. Thuscommute with every generator of the Poincare algebra. They are its two algebraically independent Casimir operators in four spacetime dimensions.