  We study optimal curvature-free inequalities of the type discovered by C.
Loewner and M. Gromov, using a generalisation of the Wirtinger inequality for
the comass. Using a model for the classifying space BS^3 built inductively out
of BS^1, we prove that the symmetric metrics of certain two-point homogeneous
manifolds turn out not to be the systolically optimal metrics on those
manifolds. We point out the unexpected role played by the exceptional Lie
algebra E_7 in systolic geometry, via the calculation of Wirtinger constants.
Using a technique of pullback with controlled systolic ratio, we calculate the
optimal systolic ratio of the quaternionic projective plane, modulo the
existence of a Joyce manifold with Spin(7) holonomy and unit middle-dimensional
Betti number.
