Self-diffusion coefficients of lithium ions and water protons (D-Li and D-H) in the glass-forming electrolyte LiCl:7H(2)O have been measured by nuclear magnetic resonance spin-echo experiments using ultrahigh static magnetic field gradients up to 184 Tm-1. The measurements were complemented by measurements of Li-7 and H-1 spin-lattice relaxation times. The data cover the temperature range from 313 K down to 173 K, i.e., 34 K above the glass transition temperature T-g=139 K. In this range D-Li and D-H change over five orders of magnitude. The self-diffusion data exhibit a strong non-Arrhenius temperature dependence which is typical for fragile glass formers. In the supercooled regime the ratio of the self-diffusion coefficients D-H/D-Li increases gradually with decreasing temperature, reflecting a decoupling of these diffusive modes. These results are discussed in relation to the behavior of the viscosity, electrical conductance and reorientational correlation time of water in this temperature range. It is found that lithium ion diffusion is closely coupled to these other transport processes, while proton diffusion begins to decouple at T<1.5 T-g. Additionally, an analysis of H-1 and Li-7 magnetic relaxation rates 1/T-1 is given. It is found that the intermolecular modes causing H-1-H-1 dipolar relaxation and Li-7 quadrupolar relaxation also decouple from the viscosity. The results are discussed in the framework of similar phenomena observed with other fragile glasses and, more specifically, of structural changes known to occur in supercooled LiCl:H2O systems.