A Complex Higher-Dimensional Lie Algebra with Real and Imaginary Structure Constants as Well as Its Decomposition

A new Lie algebra G of the Lie algebra sl(2) is constructedwith complex entries whose structure constants are real andimaginary numbers. A loop algebra ˜G corresponding to the Lie algebra G is constructed, for which it is devoted togenerating a soliton hierarchy of evolution equations under theframework of generalized zero curvature equation which is derivedfrom the compatibility of the isospectral problems expressed byHirota operators. Finally, we decompose the Lie algebra G to obtain the subalgebras G₁ and G₂. Using the G₂ and its one type of loop algebra ˜G₂, a Liouville integrable soliton hierarchy is obtained, furthermore, we obtain itsbi-Hamiltonian structure by employing the quadratic-form identity.