An electrolyzer for splitting molecular water into molecular hydrogen and molecular oxygen using electrical energy comprises an anodic half-cell with an anode and a cathodic half-cell with a cathode. The anodic half-cell and the cathodic half-cell are separated from each other by a separator. The anodic half-cell comprises an anodic electrolyte, which is in contact with the anode. The cathodic half-cell comprises a cathodic electrolyte, which is in contact with the cathode. The anodic half-cell comprises an anodic catalyst. The cathodic half-cell contains at least one cation complex for forming at least one mediator complex. The at least one cation complex is reducible to the mediator complex by taking up at least one electron at the cathode. The mediator complex is a catalytically active chemical complex for splitting the molecular water (H2O) into molecular hydrogen (H2) and hydroxide ions (OH−) while releasing at least one electron.