Thus, the contribution of \( [F, T_2]_{2b} \) to the matrix element \( \overline{H}_{ij}^{ab} \) is
$$ \begin{align*} \overline{H}_{ij}^{ab} \leftarrow \sum_{c} \left(f_c^a t_{ij}^{cb}-f_c^b t_{ij}^{ac}\right) - \sum_{k} \left(f_j^k t_{ik}^{ab}-f_i^k t_{jk}^{ab}\right) \end{align*} $$Here we used an arrow to indicate that this is just one contribution to this matrix element. We see that the derivation straight forward, but somewhat tedious. As no one likes to commute too much (neither in this example nor when going to and from work), and so we need a better approach. This is where diagramms come in handy.