Q:

rational parameterizations of Lie groups

What is a good reference for an overview of rational parameterizations of
Lie groups (like the Cayley transform for the groups of invariance of non-degenerate bilinear forms)? Is the classification of rational parameterizations known?    

A:

Concerning rational parametrizations of algebraic groups similar to the Cayley transform see this answer, which deals with Cayley maps, i.e., equivariant birational isomorphisms between an algebraic group and its Lie algebra.
If you are interested in birational parametrizations which are not necessarily equivariant, look in papers by Chernousov and Platonov, for example:  Chernousov, Vladimir I.; Platonov, Vladimir P. The rationality problem for semisimple group varieties. J. Reine Angew. Math. 504 (1998), 1–28.

