Massless longitudinal Pauli-Lubanski eigenvalues ($(W_0,W_3)=(k{j}_3,-k{j}_3)$)

With signature $(+---)$ and ${ϵ}_{0123}=+1$, a momentum $(k,0,0,k)$ and helicity $j_3$ give lower components $W_0=k{j}_3$ and $W_3=-k{j}_3$. They obey $W_{\mu }{P}^{\mu }=k(W_0+W_3)=0$. These longitudinal identities require only the momentum and rotation-projection eigenvalues, not a statement about transverse little-group translations.