  Let $A$ be an Azumaya algebra over a field. If $G$ is the group of
automorphisms of $A$ and $X$ denotes a projective homogeneous variety under
$G$, we construct in a very explicit way and under suitable hypotheses a bundle
$\mathcal{V}$ on $S$, where $S$ is a (generalized) Severi-Brauer variety
associated to $A$, and a canonical isomorphism between $X$ and a flag bundle on
$\mathcal{V}$. This allows to explicitely compute Chow groups of $X$ in terms
of the Chow groups of $S$.
