  We classify the tube domains in C^4 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 hypersurface. Using the geometry of these two hypersurfaces, we
find the automorphism groups of the domains.
