Towards a classification of homogeneous tube domains in ℂ⁴

We classify the tube domains in ℂ⁴ with affinely homogeneous base whose boundary contains a non-degenerate affinely homogeneous hypersurface. It follows that these domains are holomorphically homogeneous and amongst them, there are four new examples of unbounded homogeneous domains (that do not have bounded realisations). These domains lie to either side of a pair of Levi-indefinite hypersurfaces. Using the geometry of these two hypersurfaces, we find the automorphism groups of the domains.