Expansion of the Lie algebras and integrable couplings

We take the Lie algebra A1 as an example to illustrate a detail approach for expanding a finite dimensional Lie algebra into a higher-dimension one. Here a higher-dimension 6 × 6 matrix Lie algebra is given, which can be used to directly construct integrable couplings of the soliton integrable systems. Finally, we obtain the integrable coupling of a new integrable system and apply the quadratic-form identity to it.