With Car–Parrinello molecular dynamics simulations the elementary reaction steps of the electrolysis of bulk water are investigated. To simulate the reactions occurring near the anode and near the cathode, electrons are removed or added, respectively. The study focuses on the reactions in pure water. Effects depending on a particular electrode surface or a particular electrolyte are ignored. Under anodic conditions, the reaction continues till molecular oxygen is formed, under cathodic conditions the formation of molecular hydrogen is observed. In addition the formation of hydrogen peroxide is observed as an intermediate of the anodic reaction. The simulations demonstrate that the electrochemistry of oxygen formation without direct electrode contact can be explained by radical reactions in a solvent. These reactions may involve the intermediate formation of ions. The hydrogen formation is governed by rapid proton transfers between water molecules.