Principal bundles over the projective line

Let G be a connected reductive linear algebraic group defined over a field k and EG a principal G-bundle over the projective line satisfying the condition that EG is trivial over some k-rational point of . 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].