  We prove that in the polynomial ring $Q=\mathsf{k}[x,y,z,w]$, with
$\mathsf{k}$ an algebraically closed field of characteristic zero, all
Gorenstein homogeneous ideals $I$ such that $(x,y,z,w)^4\subseteq I \subseteq
(x,y,z,w)^2$ can be obtained by \emph{doubling} from a grade three perfect
ideal $J\subset I$ such that $Q/J$ is a locally Gorenstein ring. Moreover, a
graded minimal free resolution of the $Q$-module $Q/I$ can be completely
described in terms of a graded minimal free resolution of the $Q$-module $Q/J$
and a homogeneous embedding of a shift of the canonical module $\omega_{Q/J}$
into $Q/J$.
