Elementary triatomic reactions offer a compelling test of our understanding of the extent of electron-nuclear coupling in chemical reactions, which is neglected in the widely applied Born-Oppenheimer (BO) approximation. The BO approximation predicts that in reactions between chlorine (Cl) atoms and molecular hydrogen, the excited spin-orbit state (Cl*) should not participate to a notable extent. We report molecular beam experiments, based on hydrogen-atom Rydberg tagging detection, that reveal only a minor role of Cl*. These results are in excellent agreement with fully quantum-reactive scattering calculations based on two sets of ab initio potential energy surfaces. This study resolves a previous disagreement between theory and experiment and confirms our ability to simulate accurately chemical reactions on multiple potential energy surfaces.