We present diffusion coefficient and shear viscosity data for the Lennard-Jones fluid along nine isochores above the critical density, each involving a temperature variation of roughly two orders of magnitude. The data are analyzed with respect to the Stokes-Einstein (SE) relation, which breaks down gradually at high temperatures. This is rationalized in terms of the faet that the reduced diffusion coefficient 5 and the reduced viscosity ij are both constant along the system's lines of constant excess entropy (the isomorphs). As a consequence, Dij is a function of T /T Ref(p) in which Tis the temperature, pis the density, and T Ref(p) is the temperature as a function of the density along a reference isomorph. This allows one to successfully predict the viscosity from the diffusion coefficient in the studied region of the thermodynamic phase diagram.