Abstract
We show that on a particular class of semi-direct sums of matrix Lie algebras, component traces of the matrix product can produce bilinear forms which are non-degenerate, symmetric and invariant under the Lie product. The corresponding variational identities are called component-trace identities and provide tools in generating Hamiltonian structures of integrable couplings including the perturbation equations. An illustrative example of applying component-trace identities is given for the KdV hierarchy.