  We use the exterior and composition products of double forms together with
the alternating operator to reformulate Pontrjagin classes and all Pontrjagin
numbers in terms of the Riemannian curvature. We show that the alternating
operator is obtained by a succession of applications of the first Bianchi sum
and we prove some useful identities relating the previous four operations on
double forms. As an application, we prove that for a $k$-conformally flat
manifold of dimension $n\geq 4k$, the Pontrjagin classes $P_i$ vanish for any
$i\geq k$. Finally, we study the equality case in an inequality of Thorpe
between the Euler-Poincar\'e charateristic and the $k$-th Pontrjagin number of
a $4k$-dimensional Thorpe manifold.
