Classification of simply-transitive Levi non-degenerate hypersurfaces in ${\mathbb{C}}^3$

Holomorphically homogeneous CR real hypersurfaces $M^3\subset {\mathbb{C}}^2$ were classified by 'Elie Cartan in 1932. In the next dimension, we complete the classification of simply-transitive Levi non-degenerate hypersurfaces $M^5 \subset $\mathbb{C}$^3$ 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.