In the Hartree-Fock method for many-electron atoms, for a given eigenspace $\mathscr{E}$ of the unperturbed Hamiltonian (in a.u.) $$\hat{H}=\sum_{i=1}^Z \Big(-\frac{1}{2}\nabla^2_i-\frac{Z}{|\vec{r}_i|}+V_H(|\vec{r}_i|)\Big)$$ corresponding to the quantum numbers $\{n_1,\dots,n_Z\}\{\ell_1,\dots,\ell_Z\}\equiv\{n_i\}\{\ell_i\}$, one takes linear combinations of single-particle eigenfunctions that are antisymmetric with respect to particle exchange (i.e., Slater determinants $\{\psi_S\}$).
In $LS$-coupling scheme, $\{\psi_S\}$ are linearly combined to give common eigenstates of the total orbital and spin angular-momenta $\{\hat{L^2},\hat{S^2},\hat{L_z},\hat{S_z}\}$. In general $\{\psi_S\}$ will span a proper subspace $\mathscr{E}_S$ of $\mathscr{E}$. From what I understand, this is tantamount to saying that $\mathscr{E}_S$ can be written as a direct sum of common eigenspaces of $\{\hat{H},\hat{L^2},\hat{S^2},\hat{L_z},\hat{S_z}\}$: $$\tag{$\square$}\mathscr{E}_S = \bigoplus_k |\{n_i\}\ \{\ell_i\}\ L_k\ S_k\ \{M_{L_k}\}\ \{M_{S_k}\}\rangle$$ each of which is assigned the term symbol $^{2S+1}L$. If I understand correctly, what is the proof of $(\square)$?
