  Let $C$ be a curve and $V \to C$ an orthogonal vector bundle of rank $r$. For
$r \le 6$, the structure of $V$ can be described using tensor, symmetric and
exterior products of bundles of lower rank, essentially due to the existence of
exceptional isomorphisms between $\mathrm{Spin} (r , \mathbb{C})$ and other
groups for these $r$. We analyze these structures in detail, and in particular
use them to describe moduli spaces of orthogonal bundles. Furthermore, the
locus of isotropic vectors in $V$ defines a quadric subfibration $Q_V \subset
\mathbb{P} V$. Using familiar results on quadrics of low dimension, we exhibit
isomorphisms between isotropic Quot schemes of $V$ and certain ordinary Quot
schemes of line subbundles. In particular, for $r \le 6$ this gives a method
for enumerating the isotropic subbundles of maximal degree of a general $V$,
when there are finitely many.
