Principal bundles, quasi-abelian varieties and structure of algebraic groups

We classify principal bundles over anti-affine schemes with affine and commutative structure group. We show that this yields the classification of quasi-abelian varieties over a field k (i.e., group k-schemes G such that OG(G)=k). The interest of this result is given by the fact that the classification of smooth group k-schemes is reduced to the classification of quasi-abelian varieties and of certain affine group schemes.