 Solution

The Pauli matrices obey $[X,Z]=-2iY$ and $[Y,Z]=2iX$, and operators on different tensor factors commute. HenceAdding gives $\boxed{[X_m{X}_n+Y_m{Y}_n,Z_m+Z_n]=0}$. Each $Z_m{Z}_n$ commutes separately with both $Z_m$ and $Z_n$, so $\boxed{[Z_m{Z}_n,Z_m+Z_n]=0}$. Every interaction on sites $m,n$ also commutes with all $Z_{\ell }$ for $\ell \notin \{m,n\}$. Therefore equal $XX$ and $YY$ coefficients make their contributions cancel pair by pair, and $\boxed{[H_S,S]=0\text{ when }{\alpha }_{mn}={\beta }_{mn}\text{ for every pair}}$. This is Excitation-number conservation in XXZ spin chains: $S$ is the total spin magnetization in Pauli normalization.