Reduction theory for a rational function field

Let G be a split reductive group over a finite field F-q. Let F = F-q (t) and. let A denote the adeles of F. We show that every double coset in G(F)\G(A)/K has a representative in a maximal split torus of G. Here K is the set of integral adelic points of G. When G ranges over general linear groups this is equivalent to the assertion that any algebraic vector bundle over the projective line is isomorphic to a direct sum of line bundles.