A few subalgebras of the Lie algebra A3 and a direct approach for obtaining integrable couplings

A few subalgebras of the Lie algebra A3 are constructed from which the corresponding loop algebras can be presented. It follows that an approach for directly producing integrable couplings is given. Two examples for illustrating the way are showed. Finally, their Hamiltonian structures are worked out by using the quadratic-form identity.