An exactly solvable model for a description of the two-neutrino double-β decay transition of the Fermi type is considered. By using perturbation theory an explicit dependence of the two-neutrino double-β decay matrix element on the like-nucleon pairing, particle-particle, and particle-hole proton-neutron interactions is found by assuming a weak violation of isospin symmetry of the Hamiltonian expressed with generators of the SO(5) group. It is found that there is a dominance of double-β decay transition through a single state of the intermediate nucleus. Then, an energy-weighted sum rule connectingZ = 2 nuclei is presented and discussed. It is suggested that this sum rule can be exploited to study the residual interactions of the nuclear Hamiltonian.