Commutators of Total Spin Operators

$$\begin{eqnarray*} \vec{S} & = & \vec{S}_{(1)}+\vec{S}_{(2)} \\ \left[S_i,S_j\ri... ...k+0+0+i\hbar\epsilon_{ijk}S^{(2)}_k =i\hbar\epsilon_{ijk}S_k \\ \end{eqnarray*}$$

Q.E.D.