On $7$-dimensional Lie algebras admitting Levi-nondegenerate orbits in $\mathbb{C}^4$

A scheme for searching for holomorphically homogeneous Levi-nondegenerate real hypersurfaces in $\mathbb{C}^4$ that are not reducible to tubular manifolds and are orbits of $7$-dimensional real Lie algebras is proposed. The family of such algebras contains more than a thousand different types, but the class of homogeneous hypersurfaces under discussion turns out to be meagre. Thus, in previous (including joint) works of the author of the present article, it was shown that, out of $594$ types of solvable indecomposable Lie algebras of dimension $7$ that have $6$-dimensional nilradicals, orbits with the named properties can have no more than $104$ types of algebras. The scheme under discussion is applied to $59$ of these types, characterized by the presence of exactly two $4$-dimensional Abelian subalgebras in each algebra under consideration. It is shown that a large number of such $7$-dimensional algebras cannot have orbits with the required characteristics. At the same time, the article provides examples of the required orbits obtained using the scheme and based on simplified representations of the basis holomorphic vector fields of $6$-dimensional nilradicals of the original $7$-dimensional Lie algebras.