  We introduce characteristic classes for the spectral sequence associated to a
split short exact sequence of Hopf algebras. We show that these characteristic
classes can be seen as obstructions for the vanishing of differentials in the
spectral sequence and prove a decomposition theorem. We also interpret our
results in the settings of group and Lie algebra extensions and prove some
interesting corollaries concerning the collapse of the
(Lyndon-)Hochschild-Serre spectral sequence and the order of characteristic
classes.
