  Lee and Pandharipande studied a "double point" algebraic cobordism theory of
varieties equipped with vector bundles, and speculated that some features of
that story might extend to the case of varieties with principal G-bundles. This
note shows that this expectation holds rationally, and more generally after
inverting the torsion index of the group, for reductive G. We show that (after
inverting the torsion index) the full theory for bundles on varieties is an
extension of scalars of standard algebraic cobordism, that the theory for a
point is dual to Omega^*(BG), and describe how the theory for G compares to
that for a maximal torus T.
