Classification of simply-transitive Levi non-degenerate hypersurfaces in C^3

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 is a new coordinate-free formula for the fundamental (complexified) quartic tensor. Our final result has a unique (Levi-indefinite) non-tubular model, for which we demonstrate geometric relations to planar equi-affine geometry