It is easy to derive the commutation and anticommutation rules for the fermionic creation operators \( a_\alpha^{\dagger} \). Using the antisymmetry of the states (3) $$ \begin{equation} |\alpha_1\dots \alpha_i\dots \alpha_k\dots \alpha_n\rangle_{\mathrm{AS}} = - |\alpha_1\dots \alpha_k\dots \alpha_i\dots \alpha_n\rangle_{\mathrm{AS}} \tag{4} \end{equation} $$ we obtain $$ \begin{equation} a_{\alpha_i}^{\dagger} a_{\alpha_k}^{\dagger} = - a_{\alpha_k}^{\dagger} a_{\alpha_i}^{\dagger} \tag{5} \end{equation} $$