LL RR double insertion over the one with ${\cal I}m(A_{ii})$, essentially unaffected by triplets. This cannot be derived by the - otherwise elegant - technique based on the allowed invariants [@romstr].

When right-handed neutrinos and Higgs triplets are simultaneously present, the predicted range for the radiatively-induced $d_e$ is already sizeably excluded by the present experimental limit. This in turn is translated into a strong constraint on the imaginary part of the combination of Yukawa couplings which is relevant for the LL RR amplitude. Such a constraint could be hardly evaded even for large values of the triplet mass and unfavorable supersymmetric mass parameters. The radiatively-induced $d_\mu$ in the presence of triplets should not exceed $10^{-23}$ e cm, as was the case for the pure seesaw. However, at difference of the latter case, in the former one $d_\mu$ only mildly depends on the spectrum of right-handed neutrinos. Notice that planned searches for $d_\mu$ could constrain (depending on the triplet mass mass, of course) the imaginary part of the combination of Yukawa couplings whose absolute value could be independently constrained by the LFV decay $\tmg$.

The paper is organized as follows. In Section 2 we introduce our notations, discuss the framework and 