TITLE: Rationality of algebraic groups
QUESTION [5 upvotes]: The Cayley parametrization of $O(n),$ as in my answer to this question makes one wonder: which algebraic groups are actually rational? I am sure this is very well understood, just not by me...

REPLY [6 votes]: A reductive group $G$ over a ground field $k$ need not be a rational variety over $k$ (although the group $G_{/\ell}$ obtained by scalar extension is a rational $\ell$ variety, where $\ell$ is an algebraic closure of $k$). Maybe a good place to look concerning these matters is Merkurjev's 1996 Publ. Math. IHES paper "R-equivalence and rationality problem for semisimple adjoint classical algebraic groups".
In the intro to that paper, Merkurjev makes some interesting observations:

Chevalley showed in the 50s that there are algebraic tori over local fields which are not rational varieties.
algebraic tori over dimension <= 2 are always rational varieties; thus, any reductive
group of rank <=2 is rational. (Chevalley proved that the variety of maximal tori in a connected group is a rational variety, so the rationality of $G$ boils down to the rationality of some maximal torus of $G$).
any semisimple group over a number field which is a counterexample to weak approximation is an example of a non-rational group.

In Merkurjev's paper, you will find explicit examples of groups $G$ which are not stably rational -- a $k$-variety $X$ is stably rational if $X \times \operatorname{Aff}^d$ is a rational $k$-variety for some $d \ge 0$. 
One way to show that $G$ is not stably rational (and hence not rational) is to show that
the "group of $R$-equivalence classes" for $G$ is non-trivial;
this group of $R$-equivalence classes was introduced by Manin. Merkurjev's paper is devoted to the computation of the group of $R$-equivalence classes for semisimple, adjoint, classical groups.