The quadratic-form identity for constructing the Hamiltonian structure of integrable systems

A usual loop algebra, not necessarily the matrix form of the loop algebra (A) over tilde (n-1), is also made use of for constructing linear isospectral problems, whose compatibility conditions exhibit a zero-curvature equation from which integrable systems are derived. In order to look for the Hamiltonian structure of such integrable systems, a quadratic-form identity is created in the present paper whose special case is just the trace identity; that is, when taking the loop algebra (A) over tilde (1), the quadratic-form identity presented in this paper is completely consistent with the trace identity.