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Based on recent calculations of the self-diffusion (SD) coefficient in amorphous silicon (a-Si) by classical Molecular Dynamics simulation [M. Posselt, H. Bracht, and D. Radić, J. Appl. Phys. 131, 035102 (2022)] detailed investigations on atomic mechanisms are performed. For this purpose two Stillinger-Weber-type potentials are employed, one strongly overestimates the SD coefficient, while the other leads to values much closer to the experimental data. By taking into account the individual squared displacements (or diffusion lengths) of atoms the diffusional and vibrational contributions to the total mean squared displacement can be determined separately. It is shown that the diffusional part is not directly correlated with the concentration of coordination defects. The time-dependent distribution of squared displacements of atoms indicates that in a-Si a well-defined elemental diffusion length does not exist, in contrast to SD in the crystalline Si. The analysis of atoms with large squared displacements reveals that the mechanisms of SD in a-Si are characterized by complex rearrangements of bonds or exchange of neighbors. These are mono- and bi-directional exchanges of neighbors and neighbor replacements. Exchanges or replacements may concern up to three neighbors and may occur in relatively short periods of some ps. Bi- or mono-directional exchange or replacement of one neighbor atom happen more frequently than processes including more neighbors. A comparison of results for the two interatomic potentials shows that an increased three-body parameter only slows down the migration, but does not change the migration mechanisms fundamentally.