A higher-dimensional Lie algebra and its decomposed subalgebras

A higher-dimensional 4×44×4 matrix Lie algebra G is presented, which is devoted to setting up an isospectral Lax pair whose compatibility generates an integrable coupling system. By employing the quadratic-form identity, its Hamiltonian structure is obtained. Furthermore, a higher-dimensional Lie algebra denoted by E, is given by decomposing the Lie algebra G. If E   is decomposed into two subalgebras E1E1 and E2E2 and we require the closure between E1E1 and E2E2 under the matrix multiplication, that is, E1E2E1E2, E2E1⊂E2E2E1⊂E2, then a discrete lattice integrable coupling system is worked out. A remarkable feature of the Lie algebras G and E is used to directly construct integrable couplings.