The self-diffusion of a monatomic solute in liquid 1-octanol and n-tetradecane was investigated by means of a molecular dynamics simulation. The diffusion coefficient of a solute as small as argon is much greater than that obtained from the hydrodynamic-based Stokes-Einstein (SE) relation, as was reported experimentally. A relaxation of the memory function of a freely diffusing solute is much faster than that of the autocorrelation function of a shear stress. However, the SE behavior is recovered when the solute is spatially fixed, and the diffusion coefficient is calculated from the force-force autocorrelation function. A relaxation of the autocorrelation function of the force also follows that of shear stress. The fast diffusion of a small solute is thus ascribed to the decoupling between the structural relaxation of solvent and the solute diffusion induced by the self-motion of the solute.