Massless little-group generators ($N_1=K_1-J_2,N_2=K_2+J_1$)

For a standard null momentum $(E,0,0,E)$ in signature $(+---)$, define $K_i=M^{0i}$ and $J_i={ϵ}_{ijk}{M}^{jk}/2$. The stabilizing generators are $J_3,N_1,N_2$. The Poincare algebra gives $[J_3,N_1]=i{N}_2$, $[J_3,N_2]=-i{N}_1$ and $[N_1,N_2]=0$, the Lie algebra of planar Euclidean motions. Evaluating the Pauli-Lubanski pseudovector gives $W^2=-E^2(N_1^2+N_2^2)$.