On nondegenerate orbits of $7$-dimensional Lie algebras containing a $3$-dimensional Abelian ideal

This paper is related to the problem of describing homogeneous real hypersurfaces of multidimensional complex spaces as orbits of the action of Lie groups and algebras in these spaces. We study realizations in the form of algebras of holomorphic vector fields in $\mathbb{C}^4$ of $7$-dimensional Lie algebras containing only $3$-dimensional Abelian ideals and subalgebras. Among $594$ types of $7$-dimensional solvable indecomposable Lie algebras containing a $6$-dimensional nilradical, there are five types of such algebras. The article describes all their realizations that admit nondegenerate in the sense of Levi $7$-dimensional orbits. The presence of “simply homogeneous” orbits among the constructed hypersurfaces is shown.