On Orbits of 4-Dimensional Representations of 3-Dimensional Solvable Lie Algebras

In a 4-dimensional real space, we discuss linearly homogeneous hypersurfaces that are the orbits of 3-dimensional solvable Lie algebras. A complete list of inequivalent representations of a decomposable 3-dimensional algebra in space is given and a list of all (up to linear transformations) 3-dimensional orbits of such representations. Most of the resulting hypersurfaces have trivial affine stabilizers and are new in the problem of describing affinely homogeneous varieties.