It seems that our search of integrability runs into a dead end. Let me deliver the funeral oration. Our assumption is that there exists a basis such that when the deformation matrix is diagonal the deformation is integrable. So our strategy is to start with the defining basis and then to solve for the Lax pair. If the Lax pair exists and depends on non-trivial free parameter, we can declare the integrability. While if the non trivial Lax pair was not found, we should consider other basis. In other words, we should consider the automorphism of the Lie algebra. Since the equation is covariant under orthogonal transformation, the basis which are related by orthogonal transformation which are also the inner automorphism of the Lie algebra are equivalent. Therefore, the useful automorphism is the outer automorphism of the Lie algebra. Unfortunately, for most of the semi-simple Lie algebras the outer automorphism is trivial.
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