Massive Pauli-Lubanski Casimir ($W^2=-m^{2}s(s+1)$)

For a massive positive-energy unitary irreducible representation of the Poincaré group with signature $(+---)$, choose a rest frame. The Pauli-Lubanski pseudovector has zero time component and spatial components $m{J}_i$, up to a common sign convention. Its square is therefore $-m^2{\mathbf{J}}^2$. The SU(2) little-group representation has spin $s$ and ${\mathbf{J}}^2=s(s+1)$, proving the invariant value. The mass and spin label the massive induced representation of the Poincare double cover.