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

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