  We focus on the definition of the unitary transformation leading to an
effective second order Hamiltonian, inside degenerate eigensubspaces of the
non-perturbed Hamiltonian. We shall prove, by working out in detail the
Su-Schrieffer-Heeger Hamiltonian case, that the presence of degenerate states,
including fermions and bosons, which might seemingly pose an obstacle towards
the determination of such "Froehlich-transformed" Hamiltonian, in fact does
not: we explicitly show how degenerate states may be harmlessly included in the
treatment, as they contribute with vanishing matrix elements to the effective
Hamiltonian matrix. In such a way, one can use without difficulty the
eigenvalues of the effective Hamiltonian to describe the renormalized energies
of the real excitations in the interacting system. Our argument applies also to
few-body systems where one may not invoke the thermodynamic limit to get rid of
the "dangerous" perturbation terms.
