Principal bundles on affine space and bundles on the projective line (Q1112891)

Let k be any field and G a connected absolutely almost simple algebraic group over k. Let \(k_ s\) be the separable closure of k. Then it is shown that if P is a G-bundle on the affine space \({\mathbb{A}}^ n\), trivial over \(k_ s\) and over a k-point of \({\mathbb{A}}^ n\) and G is \textit{isotropic} over k, then P is trivial. On the other hand it is shown that if G is anisotropic over k and satisfies a further condition (known to hold in many cases), then there are G-bundles P on \({\mathbb{A}}^ 2\) trivial over \(k_ s\) and over a k- point such that P does not admit a reduction of structure group a connected subgroup of G.