A non-split torsor with trivial fixed point obstruction

Let G be a linear algebraic group and X be an irreducible algebraic variety with a generically free G-action, all defined over an algebraically closed base field of characteristic zero. It is well known that X can be viewed as a G-torsor, representing a class [X] in H1(K,G), where K is the field of G-invariant rational functions on X. We have previously shown that if X has a smooth H-fixed point for some non-toral diagonalizable subgroup H of G then [X]≠1. It is natural to ask if the converse is true, assuming G is connected and X is projective and smooth. In this note we show that the answer is “no.”