  We translate into the double forms formalism the basic identities of Greub
and Greub-Vanstone that were obtained in the mixed exterior algebra. In
particular, we introduce a second product in the space of double forms, namely
the composition product, which provides this space with a second associative
algebra structure. The composition product interacts with the exterior product
of double forms; the resulting relations provide simple alternative proofs to
some classical linear algebra identities as well as to recent results in the
exterior algebra of double forms.\\ We define a refinement of the notion of
pure curvature of Maillot and we use one of the basic identities to prove that
if a Riemannian $n$-manifold has $k$-pure curvature and $n\geq 4k$ then its
Pontrjagin class of degree $4k$ vanishes.
