  We observe that, in the eta-periodic motivic stable homotopy category, odd
rank vector bundles behave to some extent as if they had a nowhere vanishing
section. We discuss some consequences concerning SLc-orientations of motivic
ring spectra, and the etale classifying spaces of certain algebraic groups. In
particular, we compute the classifying spaces of diagonalisable groups in the
eta-periodic motivic stable homotopy category.
