  Let $k$ be a field of characteristic $0$. Using the method of idealization,
we show that there is a non-Koszul, quadratic, Artinian, Gorenstein, standard
graded $k$-algebra of regularity $3$ and codimension $8$, answering a question
of Mastroeni, Schenck, and Stillman. We also show that this example is minimal
in the sense that no other idealization that is non-Koszul, quadratic,
Artinian, Gorenstein algebra, with regularity $3$ has smaller codimension.
  We also construct an infinite family of graded, quadratic, Artinian,
Gorenstein algebras $A_m$, indexed by an integer $m \ge 2$, with the following
properties: (1) there are minimal first syzygies of the defining ideal in
degree $m+2$, (2) for $m \ge 3$, $A_m$ is not Koszul, (3) for $m \ge 7$, the
Hilbert function of $A_m$ is not unimodal, and thus (4) for $m \ge 7$, $A_m$
does not satisfy the weak or strong Lefschetz properties. In particular, the
subadditivity property fails for quadratic Gorenstein ideals.
  Finally, we show that the idealization of a construction of Roos yields
non-Koszul quadratic Gorenstein algebras such that the residue field $k$ has a
linear resolution for precisely $\alpha$ steps for any integer $\alpha \ge 2$.
Thus there is no finite test for the Koszul property even for quadratic
Gorenstein algebras.
