Classification of Simply-Transitive Levi Non-Degenerate Hypersurfaces in ?3

Abstract Holomorphically homogeneous Cauchy–Riemann (CR) real hypersurfaces $M^3 \subset \mathbb{C}^2$ were classified by Élie Cartan in 1932. In the next dimension, we complete classification of simply-transitive Levi non-degenerate $M^5 \mathbb{C}^3$ using a novel Lie algebraic approach independent any earlier classifications abstract algebras. Central to our is new coordinate-free formula for fundamental (complexified) quartic tensor. Our final result has unique (Levi-indefinite) non-tubular model, which demonstrate geometric relations planar equi-affine geometry.