  In this paper a list of $R$-matrices on a certain coupled Lie algebra is
obtained. With one of these $R$-matrices, we construct infinitely many
bi-Hamiltonian structures for each of the two-component BKP and the Toda
lattice hierarchies. We also show that, when such two hierarchies are reduced
to their subhierarchies, these bi-Hamiltonian structures are reduced
correspondingly.
