The dynamics of Li^{+} transport in polyethylene oxide (PEO) and lithium bis(trifluoromethanesulfonyl)imde mixtures are investigated by combining neutron spin-echo (NSE) and dielectric spectroscopy with molecular dynamics (MD) simulations. The results are summarized in a relaxation time map covering wide ranges of temperature and time. The temperature dependence of the dc conductivity and the dielectric α relaxation time is found to be identical, indicating a strong coupling between both. The relaxation times obtained from the NSE measurements at 0.05 Å^{-1}<q<0.2 Å^{-1} are of similar magnitude as the relaxation time of Li^{+} predicted by MD simulation. Our results suggest that the characteristic live times of the ions within the oxygen cages are mainly determined by the α relaxation that corresponds to local segmental motions of polymers, to a much lesser extent by the main chain relaxation, and not at all by the β relaxation or other faster processes. It is the first time decisive experimental evidence for a microscopic picture of the Li ion transportation process is shown in which the PEO chain forms EO cages over several monomer units and the Li ion "jump" from cage to cage. The role of the backbone of the polymer is discussed and contributes signifcantly to the Li ion transportation process. Moreover, detailed characteristic length and time scales of the Li^{+} transport process in this polymer electrolyte are identified and interpreted.