This work presents the results of the structural characterization of LaSrGa1-xMgxO4-delta oxides with x = 0.0-0.2 (LSGM'); these oxides with layered K2NiF4-type structure are potential electrolytes of oxygen ions with applications in solid oxide fuel cells (SOFCs). Neutron powder diffraction (NPD), in complement with electrical characterization by AC impedance spectroscopy, enabled the determination of the pathways and mechanisms for diffusion of oxygen ions within the crystalline structure. It was found that the presence of Mg in the structure promotes a non-stoichiometry of oxygen vacancies, which significantly improves the ionic conduction properties by three orders of magnitude. Structural refinements from NPD data in complement with Fourier maps and bond-valence energy landscape (BVEL) calculations, revealed that the oxygen diffusion occurs through the NaCl layer using the 16n crystallographic site of the I4/mmm space group. The oxygen vacancies are concentrated in the 4c sites (O2 oxygen atoms) with (delta) of 0.008 and 0.08 for x = 0.1 and x = 0.2, respectively. The dominant oxygen-ion electrical conductivity increased from order mu S cm(-1) to mS cm(-1) at 900 degrees C.