Given an exponential Lie group , we show that the constructions of B. Currey, 1992, go through for a less restrictive choice of the Jordan-Hölder basis. Thus we obtain a stratification of into -invariant algebraic subsets, and for each such subset , an explicit cross-section for coadjoint orbits in , so that each pair behaves predictably under the associated restriction maps on . The cross-section mapping is explicitly shown to be real analytic. The associated Vergne polarizations are not necessarily real even in the nilpotent case, and vary rationally with . For each , algebras and of polarized and quantizable functions, respectively, are defined in a natural and intrinsic way.

Now let be the dimension of coadjoint orbits in . An explicit algorithm is given for the construction of complex-valued real analytic functions and such that on each coadjoint orbit in , the canonical 2-form is given by . The functions belong to , and the functions belong to . The associated geometric polarization on each orbit coincides with the complex Vergne polarization, and a global Darboux chart on is obtained in a simple way from the coordinate functions (restricted to ). Finally, the linear evaluation functions are shown to be quantizable as well