Spin commutation relations ($[S_i,S_j]=i\hslash {ϵ}_{ijk}{S}_k$)

The components of spin angular momentum satisfy $[S_i,S_j]=i\hslash {ϵ}_{ijk}{S}_k$. In units $\hslash =1$, the spin ladder operators obey $[S^{+},S^{-}]=2S^z$ and $[S^z,S^{\pm }]=\pm S^{\pm }$.