Right now we use a spin representation where we map spins onto electron pairs. This is convenient since it maps spin-models onto seniority-zero Hamiltonians/wavefunctions. It is not convenient in the sense that the $\hat{S}_x$ and $\hat{S}_y$ operators break particle-number symmetry, so a general XYZ Heisenberg model is inaccessible.
There is an alternative approach. We can directly model the spin on each orbital as $Sk^+ = a{k\alpha}^\dagger a_{k\beta}$ converting a down-spin electron in spatial orbital $k$ to an up-spin electron in spatial orbital $k$ and annhilating the wavefunction if the orbital is empty or has a up-spin electron already in it. Similarly, $Sk^- = a{k\beta}^\dagger a_{k\alpha}$ changes an up-spin electron in orbital $k$ to a down-spin electron, and otherwise gives zero. $S^Zk=\tfrac{1}{2}\left(a{k\alpha}^\dagger a{k\alpha}-a{k\beta}^\dagger a_{k\beta} \right)$ gives $\pm \tfrac{1}{2}$ depending on whether the $k$-th orbital has a spin-up or spin-down electron.
This is a legitimate representation of the spin algebra because
The interesting state then is the maximum seniority state where every spatial orbital is singly-occupied. However, it seems that now a fully general XYZ model is allowable, because terms like $S_k^+ \pm Sl^-$ are now allowed. The key symmetry to maintain now is the maximum-seniority-sector, noting that we need to keep track of cases with different numbers of $N\alpha$ and $N_{\beta}$ electrons (so when it comes time to solve the Schrödinger equation, it may be easiest to support this in the generalized framework).
Right now we use a spin representation where we map spins onto electron pairs. This is convenient since it maps spin-models onto seniority-zero Hamiltonians/wavefunctions. It is not convenient in the sense that the $\hat{S}_x$ and $\hat{S}_y$ operators break particle-number symmetry, so a general XYZ Heisenberg model is inaccessible.
There is an alternative approach. We can directly model the spin on each orbital as $Sk^+ = a{k\alpha}^\dagger a_{k\beta}$ converting a down-spin electron in spatial orbital $k$ to an up-spin electron in spatial orbital $k$ and annhilating the wavefunction if the orbital is empty or has a up-spin electron already in it. Similarly, $Sk^- = a{k\beta}^\dagger a_{k\alpha}$ changes an up-spin electron in orbital $k$ to a down-spin electron, and otherwise gives zero. $S^Zk=\tfrac{1}{2}\left(a{k\alpha}^\dagger a{k\alpha}-a{k\beta}^\dagger a_{k\beta} \right)$ gives $\pm \tfrac{1}{2}$ depending on whether the $k$-th orbital has a spin-up or spin-down electron.
This is a legitimate representation of the spin algebra because
$$ [S_k^+ , Sl^-] = 2\delta{kl} S_k^Z $$
$$ [S_k^Z, Sl^{\pm}] = \pm \delta{kl} S_k^{\pm} $$
The interesting state then is the maximum seniority state where every spatial orbital is singly-occupied. However, it seems that now a fully general XYZ model is allowable, because terms like $S_k^+ \pm Sl^-$ are now allowed. The key symmetry to maintain now is the maximum-seniority-sector, noting that we need to keep track of cases with different numbers of $N\alpha$ and $N_{\beta}$ electrons (so when it comes time to solve the Schrödinger equation, it may be easiest to support this in the generalized framework).