Almost positive curvature on the Gromoll-Meyer sphere

Gromoll and Meyer have represented a certain exotic 7-sphere ${\mathrm{\Sigma }}^7$ as a biquotient of the Lie group $G=Sp(2)$. We show for a 2-parameter family of left invariant metrics on $G$ that the induced metric on ${\mathrm{\Sigma }}^7$ has strictly positive sectional curvature at all points outside four subvarieties of codimension $\ge 1$ which we describe explicitly.