  Let $\mathcal{O}$ be an order of index $m$ in the maximal order of a
quadratic number field $k=\mathbb{Q}(\sqrt{d})$. Let
$\underline{\mathbf{O}}_{d,m}$ be the orthogonal $\mathbb{Z}$-group of the
associated norm form $q_{d,m}$. We describe the structure of the pointed set
$H^1_{\mathrm{fl}}(\mathbb{Z},\underline{\mathbf{O}}_{d,m})$, which classifies
quadratic forms isomorphic (properly or improperly) to $q_{d,m}$ in the flat
topology. Gauss classified quadratic forms of fundamental discriminant and
showed that the composition of any binary $\mathbb{Z}$-form of discriminant
$\Delta_k$ with itself belongs to the principal genus. Using cohomological
language, we extend these results to forms of certain non-fundamental
discriminants.
