  Let $(R,\mathfrak{m},\Bbbk)$ be a regular local ring of dimension 3. Let $I$
be a Gorenstein ideal of $R$ of grade 3. Buchsbaum and Eisenbud proved that
there is a skew-symmetric matrix of odd size such that $I$ is generated by the
sub-maximal pfaffians of this matrix. Let $J$ be the ideal obtained by
multiplying some of the pfaffian generators of $I$ by $\mathfrak{m}$; we say
that $J$ is a trimming of $I$. Building on a recent paper of Vandebogert, we
construct an explicit free resolution of $R/J$ and compute a partial DG algebra
structure on this resolution. We provide the full DG algebra structure in the
appendix. We use the products on this resolution to study the Tor algebra of
such trimmed ideals and we use the information obtained to prove that recent
conjectures of Christensen, Veliche and Weyman on ideals of class $\mathbf{G}$
hold true in our context. Furthermore, we address the realizability question
for ideals of class $\mathbf{G}$.
