  We show that Artin-Schelter regularity of a $\mathbb{Z}$-graded algebra can
be examined by its associated $\mathbb{Z}^r$-graded algebra. We prove that
there is exactly one class of four-dimensional Artin-Schelter regular algebras
with two generators of degree one in the Jordan case. This class is strongly
noetherian, Auslander regular, and Cohen-Macaulay. Their automorphisms and
point modules are described.
