6 JONATHAN BREZIN relationshi p between A and S is explained in [l]. 3. The ad-algebrai c hull: T h e o r e m 1: Ther e exists a (unique) Lie group G for each solvable Lie grou p _S such that (1) J5 i s a n o r m a l s u b g r o u p of G . (2) Ad G is the algebrai c hull of Ad S . (3) G and S have the sam e center . Proo f (L. Auslander) : Let _T be a m a x i m a l toru s in the algebrai c hull of Ad(S). Becaus e the automorphis m group of S is algebrai c and contains Ad(S), T consist s of automorphism s of S. S being simply connected, we can view T as a group of automorphism s of S . Thus we can for m the s e m i - d i r e c t product S . Let Z be a closed subgroup of the cente r of T.S complementar y to the center of i . Then G is the uni - v e r s a l covering group of (T-S)/ Z . Fo r further details , see T h e o r e m 2.1 of [1]. QED. We shall'refe r to G as the ad-algebrai c hull of S . G has the following p r o p e r t i e s : (1) [ G , G ] = [ S , S ] . (2) G is almos t algebraic , and the n i l - r a d i c a l of G is the sam e as the n i l - r a d i c a l of the almos t algebrai c hull of S_ . (3) Let F be a connected subgroup of S_ that contains [S_, S], Then Ad G is algebraic .
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