= Solution
Use $[a_i,a_j^\dagger]=\delta_{ij}$ and define $E_{ij}=a_i^\dagger a_j$. A direct multiplication gives
$$
[E_{ij},E_{kl}]=\delta_{jk}E_{il}-\delta_{il}E_{kj}.
$$
With $S_+=\hbar a_\uparrow^\dagger a_\downarrow$, $S_-=\hbar a_\downarrow^\dagger a_\uparrow$, this implies $[S_+,S_-]=2\hbar S_z$ and $[S_z,S_\pm]=\pm\hbar S_\pm$. Resolving the raising and lowering combinations gives
$$
\boxed{[S_i,S_j]=i\hbar\varepsilon_{ijk}S_k}.
$$
These operators realize the <Schwinger boson representation> of <angular momentum>. Each commutes with $\widehat N=n_\uparrow+n_\downarrow$, so fixed total occupation is an invariant subspace.
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