Here we exploited the antisymmetry \( t_{ij}^{ab} = -t_{ji}^{ab} = -t_{ij}^{ba} = t_{ji}^{ba} \) in the last step. Using \( a^\dagger_q a^\dagger_b a_j a_i = -a^\dagger_b a^\dagger_q a_j a_i \) and \( a^\dagger_a a^\dagger_b a_p a_i = a^\dagger_a a^\dagger_b a_i a_p \), we can make the expression manifest antisymmetric, i.e.
$$ \begin{align*} [F, T_2]_{2b} &= \frac{1}{4}\sum_{qbij}\left[\sum_{a} \left(f_a^q t_{ij}^{ab}-f_a^b t_{ij}^{qa}\right)\right]a^\dagger_q a^\dagger_b a_j a_i \\ &- \frac{1}{4}\sum_{pabi}\left[\sum_{j} \left(f_p^j t_{ij}^{ab}-f_i^j t_{pj}^{ab}\right)\right]a^\dagger_a a^\dagger_b a_p a_i . \end{align*} $$