We calculate the probabilities of slepton and photino production via a charged lepton decay induced by an arbitrary electromagnetic wave or by a constant homogeneous external field, F(munu). This allows us to estimate the domains where this reaction could be significant. Comparing the process l --> l(gamma) with the synchrotron radiation we conclude that for all values of the background field parameter chi = em(l)-3 square-root -(F(munup(nu))2, with p(nu) being the lepton's momentum, the following estimate holds: GAMMA(l --> l(gamma))/GAMMA(l --> l(gamma)) less-than-or-equal-to 1/28. We also study the impact of the intense field on radiative corrections to the lepton's anomalous magnetic moment. It is found that, when chi exceeds the cube of the masses' ratios, (M(l)/m(l))3, (M(gamma)/m(l))3, the sparticle's contribution to the anomalous magnetic moment is of the same order as the QED correction. In this region the magnitude of the anomalous magnetic moment of the lepton is very sensitive to the masses of the slepton and the photino.