= Solution
The <Pauli matrices> obey $[X,Z]=-2iY$ and $[Y,Z]=2iX$, and operators on different tensor factors commute. Hence
$$
[X_mX_n,Z_m+Z_n]=-2i(Y_mX_n+X_mY_n),\qquad [Y_mY_n,Z_m+Z_n]=2i(X_mY_n+Y_mX_n).
$$
Adding gives $\boxed{[X_mX_n+Y_mY_n,Z_m+Z_n]=0}$. Each $Z_mZ_n$ commutes separately with both $Z_m$ and $Z_n$, so $\boxed{[Z_mZ_n,Z_m+Z_n]=0}$. 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.
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