Interactions between the composition and exterior products of double forms and applications

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.