An elementary proof of the structure theorem for connected solvable affine algebraic groups (Q1122657)

The author gives an elementary proof of the basic structure theorem for connected solvable affine algebraic groups G over an algebraically closed field k. First, he shows that for a semi-simple element s of G, the centralizer \(Z_ G(s)\) of s in G is connected and \(G=G_ u\cdot Z_ G(s)\), where \(G_ u\) is the closed subgroup of G consisting of all unipotent elements of G. Then, he can easily show that \(G=G_ u\rtimes T\), where T is maximal torus of G and that all maximal tori of G are conjugate. In his proof, he avoids the use of quotients and Lie algebras of affine groups but uses the Lie-Kolchin theorem, Chevalley's theorem, the existence and uniqueness of the Jordan decomposition and some other elementary facts.