  We continue the analysis, started by Abreu, McDuff and Anjos, of the topology
of the group of symplectomorphisms of $S^2 \times S^2$ when the ratio of the
areas of the two spheres lies in the interval (1,2]. We express the group, up
to homotopy, as the amalgam of certain of its compact Lie subgroups. We use
this to compute the homotopy type of the classifying space of the group of
symplectomorphisms and the corresponding ring of characteristic classes for
symplectic fibrations.
