  We discuss the relations between the violation of the CHSH Bell inequality
for systems of two qubits on the one side and entanglement of formation, local
filtering operations, and the entropy and purity on the other. We calculate the
extremal Bell violations for a given amount of entanglement of formation and
characterize the respective states, which turn out to have extremal properties
also with respect to the entropy, purity and several entanglement monotones.
The optimal local filtering operations leading to the maximal Bell violation
for a given state are provided and the special role of the resulting Bell
diagonal states in the context of Bell inequalities is discussed.
