We have used the fact that \( \sigma ^{2}_{x} = \sigma ^{2}_{y}=\sigma ^{2}_{z}=1 \). When \( k \neq k' \) the operators commute and cancel. Thus $$ \sum_{f} \left[B_{fi}(F_{-}) - B_{fi}(F_{+}) \right] = (N_{i}-Z_{i}), $$ and $$ \sum_{f} \left[ B_{fi}(GT_{-}) - B_{fi}(GT_{+}) \right] = 3(N_{i}-Z_{i}). $$

The sum-rule for the Fermi matrix elements applies even when isospin is not conserved.