Reduction theory for a rational function field

Let $G$ be a split reductive group over a finite field $\text{Fq}$. Let $F=\text{Fq}(t)$ and let $\text{A}$ denote the ad`eles of $F$. We show that every double coset in $G(F)\text{bsl}G(\text{A})\mathrm{/}K$ has a representative in a maximal split torus of $G$. Here $K$ is the set of integral ad`elic 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.