The Lancet

We classify all (locally) homogeneous Levi non-degenerate real hypersurfaces in $\mathbb{C}^3$ with symmetry algebra of dimension $\geq 6$.

We establish the submaximal symmetry dimension for Riemannian and Lorentzian conformal structures. The proof is based on enumerating all subalgebras of orthogonal Lie algebras of sufficiently large dimension and verifying if they stabilize a non-zero Weyl tensor up to scale. Our main technical tools include Dynkin's classification of maximal subalgebras in complex simple Lie algebras, a theorem of Mostow, and Kostant's Bott-Borel-Weil theorem.