Various representations of matrix Lie algebras related to homogeneous surfaces

We construct a $3$-parameter family of real homogeneous hypersurfaces in a $3$-dimensional complex space. This family generalizes several examples that were published earlier. It contains both Levi nondegenerate surfaces (strictly pseudoconvex and indefinite ones) and surfaces with degenerate Levi form.
Unlike the known cumbersome descriptions of matrix algebras corresponding to the surfaces under consideration, we propose an upper triangular representation of these algebras with simple special bases. We show that all affinely homogeneous surfaces of the constructed family are algebraic ones of degree $1,2,3,4$, or $6$.