In [22], a class of four-dimensional, quadratic, Artin-Schelter regular algebras was introduced, whose point scheme is the graph of an automorphism of a nonsingular quadric in P3. These algebras are the first examples of quadratic Artin-Schelter regular algebras whose defining relations are not determined by the point scheme and, hence, not determined by the algebraic data obtained from the point modules. In this paper, we study these algebras via their line modules. In particular, the set of lines in P3that correspond to left line modules is not the set of lines in P3that correspond to right line modules. Our analysis focuses on a distinguished memberRλof this class of algebras, whereRλis a twist by a twisting system of the other algebras. We prove thatRλis a finite module over its center and that its central Proj is a smooth quadric inP4.