The hermitian conjugate of \( a_\alpha^{\dagger} \) is $$ \begin{equation} a_{\alpha} = ( a_{\alpha}^{\dagger} )^{\dagger} \tag{9} \end{equation} $$ If we take the hermitian conjugate of Eq. (8), we arrive at $$ \begin{equation} \{a_{\alpha},a_{\beta}\} = 0 \tag{10} \end{equation} $$