It has also been shown that the algebraic dynamics might be generalized from the linear dynamic system with a semi-simple Lie algebra to that with a general Lie algebra.

When the set of roots for a semi-simple Lie algebra have been selected appropriately, the scalar product of the weights of the elementary representation can be calculated.

Using the notion of coefficient matrix and maximal element. We prove that the Lie algebra is semi-simple and it has no abelian two dimensional subalgebra.