Holomorphically homogeneous Cauchy–Riemann (CR) real hypersurfaces M³⊂C² were classified by Élie Cartan in 1932. In the next dimension, we complete the classification of simply-transitive Levi non-degenerate hypersurfaces M⁵⊂C³ using a novel Lie algebraic approach independent of any earlier classifications of abstract Lie algebras. Central to our approach ...