A theoretical study of the relaxation of an excited impurity center with strong E × e-type Jahn-Teller effect, caused by the emission of phonons to the bulk, is presented. The dependence of the passing the system through the conical intersection of the potential surface on the momentum of the pseudorotation is figured out. An analytical description of the quantum states of the conical intersection (Slonczewski resonances) is given. It is found that for realistic vibronic interactions with phonons, the characteristic time of the energy loss is several tenths of mean periods of phonons, i.e., it is in the picosecond range. It is also found that there is a finite probability of the speeding-up of the pseudorotation of the system at the intermediate stage of relaxation. In particular, this probability increases close to the Slonczewski resonances. During the relaxation, the system may change the direction of the pseudomoment; the probability of such a change also increases near the resonances.