Semi-simple splittings for solvable Lie groups and polynomial structures

In this paper we construct a semi-simple splitting for all connected and si mply connected solvable Lie groups G. Such a semi-simple splitting <(G )ove r bar> is itself a connected and simply connected solvable Lie group, conta ining G, and moreover, <(G )over bar> splits over its nilradical. The const ruction we present is the continuous analogue of a similar construction for polycyclic groups, due to D. Segal. Finally we use this semi-simple splitt ing to show that any G admits a polynomial structure of degree dim(G).