Two pairs of Lie algebras and the integrable couplings as well as the Hamiltonian structure of the Yang hierarchy

Two types of Lie algebras, which are the subalgebras of the Lie algebra A2, A3 respectively, are presented. The resulting loop algebras are following. As their applications, two different integrable couplings of the Yang hierarchy are obtained, called them the double integrable couplings. The Hamiltonian structure of one of them is worked out by a proper linear isomorphic transformation and the quadratic-form identity.