Artinian Gorenstein algebras of embedding dimension four and socle degree three

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 $Q\mathrm{/}J$ is a locally Gorenstein ring. Moreover, a graded minimal free resolution of the $Q$-module $Q\mathrm{/}I$ can be completely described in terms of a graded minimal free resolution of the $Q$-module $Q\mathrm{/}J$ and a homogeneous embedding of a shift of the canonical module ${\omega }_{Q\mathrm{/}J}$ into $Q\mathrm{/}J$.