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.