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abelian According to theorem alge algebra f t algebra g algebras with given automor called Cartan subalgebras coincides commutative semi-regular subalgebras commutative subalgebra complex numbers contained corresponding dimension direct sum element g equations finite number follows fundamental subspaces gebra given kernel group of automorphisms group of linear Hence induces inner automorphism inner conjugates invariant relative inverse image isomorphic Jordan normal form leaving SI invariant Let g Let us decompose Let us denote Levi's theorem Lie groups linear group linear subspace linear trans linear transformation maximal commutative semi-regular maximal nilpotent subalgebras maximal semi-regular subalgebra maximal solvable modulo nilpotent algebra nilpotent elements nilpotent group normal decomposition obtain permutable polynomials properties reduced regular representation semi-direct sum semi-regular elements semi-simple algebras solvable group SOLVABLE LIE ALGEBRAS solvable splittable algebra splittable solvable algebra splitting of g subal subgroup table algebra theorem 13 theorem 9 theory of invariants uniquely vector