Algebraic identity for the Schouten tensor and bi-Hamiltonian systems

A (0,30,3)-tensor TijkTijk is introduced in an invariant form. Algebraic identities are derived that connect the Schouten (2,12,1)-tensor Sijk and tensor TijkTijk with the Nijenhuis tensor Nikj. Applications to the bi-Hamiltonian dynamical systems are presented.