  We have revisited the computations of the flavor violating leptonic decays of
the $\tau$ and $\mu$ leptons into three lighter charged leptons in the Standard
Model with non-vanishing neutrino masses. We were driven by a claimed
unnaturally large branching ratio predicted for the $\tau^-\to \mu^- \ell^+
\ell^-$ ($\ell=\mu, e$) decays \cite{Pham:1998fq}, which was at odds with the
corresponding predictions for the $\mu^-\to e^- e^-e^+$ processes
\cite{Petcov:1976ff}. In contrast with the prediction in \cite{Pham:1998fq},
our results are strongly suppressed and in good agreement with the
approximation done in ref.~\cite{Petcov:1976ff}, where masses and momenta of
the external particles were neglected in order to deal with the loop integrals.
However, as a result of keeping external momenta and masses in the computation
of the dominant penguin and box diagrams- we even find slightly smaller
branching fractions. Therefore, we confirm that any future observation of such
processes would be an unambiguous manifestation of new physics beyond the
Standard Model
