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 \(Q/J\) 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 \(Q/J\).