${\cal I}m(A_{ii})$ shows up at $O(t_3^4)$, through ${\cal I}m(y_e N [N,E] N)_{ii}$. On the other hand, when also Yukawa interaction of the $SU(5)$ triplets are present, eq. (\[Ae\_su5\]) shows that, at first order in $t_T \equiv 1/(4 \pi)^2$ $\ln(M_{Pl}/ M_T)$, $A_e$ can be proportional to $y_e E, y_e N$ but also to $U^* y_e$, where $U \equiv y_u^\dagger y_u$. The latter have real diagonal elements but allow at $O(t_T^2)$ for a combination with diagonal imaginary part, namely $(U^* y_e N)_{ii}$.

Accordingly, in order to evaluate the actual coefficients in front of the products of Yukawa coupling matrices which are likely to have phases, we solve the RGE for the soft parameters by a Taylor expansion in the small parameters $t_{if}=1/(4 \pi)^2 \ln(Q_i/Q_f)$ associated to the intervals between the successive decoupling thresholds of the various heavy states. For instance, by integrating the RGE for $A_e$ in the case of $SU(5)$ plus seesaw, a non vanishing coefficient is obtained for the combination ${\cal I}m(U^* y_e N )_{ii}$ at $O(t_T^2)$. Now, lepton FV transitions and CP phases are naturally defined in the LF basis where $y_e$ and the Majorana masses $M_R$ are diagonal and real. This basis is not invariant under the RGE 