Principal bundles over the projective line

AbstractLet G be a connected reductive linear algebraic group defined over a field k and EG a principal G-bundle over the projective line Pk1 satisfying the condition that EG is trivial over some k-rational point of Pk1. If the field k is algebraically closed, then it is known that the principal G-bundle EG admits a reduction of structure group to the multiplicative group Gm. We prove this for arbitrary k. This extends the results of Harder (1968) [10] and Mehta and Subramanian (2002) [14]