 Solution

Use $[a_i,a_j^{†}]={\delta }_{ij}$ and define $E_{ij}=a_i^{†}{a}_j$. A direct multiplication givesWith $S_{+}=\hslash a_{\uparrow }^{†}{a}_{\downarrow }$, $S_{-}=\hslash a_{\downarrow }^{†}{a}_{\uparrow }$, this implies $[S_{+},S_{-}]=2\hslash S_z$ and $[S_z,S_{\pm }]=\pm \hslash S_{\pm }$. Resolving the raising and lowering combinations givesThese operators realize the Schwinger boson representation of angular momentum. Each commutes with $\hat{N}=n_{\uparrow }+n_{\downarrow }$, so fixed total occupation is an invariant subspace.