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?
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.
In this answer, we view rational parameterizations of Lie groups as approximations of the exponential map from the Lie algebra to the Lie group. Here are some references that develop this viewpoint further.
- E. Celedoni and A. Iserles. Approximating the Exponential From a Lie Algebra to a Lie Group. Mathematics of Computation (2000) Volume 69, Number 232, Pages 1457–1480.
- C. Moler and C. Van Loan. Nineteen Dubious Ways to Compute the Exponential of a Matrix. SIAM Review (2003) Vol. 45, No. 1, 3-49. See Method 2 (of 19) and references therein.
As discussed in the introduction of Ref 1, rational approximations of the matrix exponential do not in general lie on the Lie group but, for some Lie groups (e.g., quadratic Lie groups) they do.