ag.algebraic geometry - any linear algebraic group rational?

This is really just Pete Clark's answer -- the new bit is to note that
the Levi decomposition isn't needed.

Let G be a (reduced, connected) linear algebraic group over an
alg. closed k, and let R be the unipotent radical of G. Choose a Borel
group B of G with unipotent radical U < B (so R < U).

There is a dense B-orbit ("big cell") V in G/B which is a rational
variety.

Since U and R are (split) unipotent, [Springer, LAG 14.2.6] shows that there
is a section $s:U/R to U$ to the natural projection $U to U/R$.

If $f:G to G/B$ is the quotient mapping, using $s$ you can find a "local
section" of $f$ over the big cell V.

This show that $f$ is a locally trivial B-bundle (in the Zariski
topology) and in particular $f^{-1}(V)$ is an open subvariety of G
isomorphic to the rational variety V x B.