The invention relates to a method, an electrolyte membrane, and a corresponding electrolysis cell or an electrolysis stack for producing hydrogen and oxygen from water vapor using electric energy and/or a corresponding fuel cell or a fuel cell stack in order to produce electric energy using hydrogen and oxygen by means of a redox reaction of lithiated iron oxide iron which is dissolved in a liquid alkali carbonate salt. The membrane for splitting water vapor into hydrogen and oxygen consists, in the embodiment according to the invention, of a novel lithiated iron oxide electrolyte which is dissolved in a liquid alkali carbonate salt mixture, generally also referred to as a carbonate melt, which includes lithium carbonate among others. The electrolyte and the liquid carbonate salt are bonded in a heat-resistant non-conductive matrix, for example consisting of lithium aluminate LiAlO.sub.2 and/or another heat-resistant material with a capillary effect.