Both \( \hat{H}_0 \) and \( \hat{H}_I \) are invariant under all possible permutations of any two particles and hence commute with \( \hat{A} \) $$ \begin{equation} [H_0,\hat{A}] = [H_I,\hat{A}] = 0. \tag{5} \end{equation} $$ Furthermore, \( \hat{A} \) satisfies $$ \begin{equation} \hat{A}^2 = \hat{A}, \tag{6} \end{equation} $$ since every permutation of the Slater determinant reproduces it.