  We study double charge-exchange phonon states in neutron-rich nuclei, in
particular the double isobaric analog states and the double Gamow-Teller
excitations, induced by the double isospin operator $\sum_{i,j=1}^At_-(i)
t_-(j)$ and spin-isospin operator $\sum_{i,j=1}^A{\sigma}(i) t_-(i){\sigma}(j)
t_-(j)$, respectively. We employ quartic commutator relations to evaluate the
average energies $E_{\rm DIAS} - 2E_{\rm IAS}$ and $E_{\rm DGTR}-E_{\rm DIAS} -
2(E_{\rm GTR}-E_{\rm IAS})$, and conventional double commutator relations to
evaluate the average energies of $E_{\rm GTR}-E_{\rm IAS}$ and $E_{\rm IAS}$.
We have found that the corrections due to quartic commutators follow the
approximate laws: $E_{\rm DIAS} - 2E_{\rm IAS}\approx \frac{3}{2} A^{-1/3}$ MeV
and $E_{\rm DGTR}-E_{\rm DIAS} - 2(E_{\rm GTR}-E_{\rm IAS})\approx 16 A^{-1}$
MeV. While the former is dominated by direct Coulomb effects, since Coulomb
exchange cancels out to some extent with isospin symmetry breaking
contributions originated form the nuclear strong force, the latter is sensitive
to the difference in strength between the spin and spin-isospin chanels of the
strong interaction.
