A compact Riemannian homogeneous space G/H, with a bi--invariant orthogonal
decomposition g=h+m is called positively
curved for commuting pairs, if the sectional curvature vanishes for any tangent
plane in TeH​(G/H) spanned by a linearly independent commuting pair in
m. In this paper,we will prove that on the coset space
Sp(2)/U(1), in which U(1) corresponds to a short
root, admits positively curved metrics for commuting pairs. B. Wilking recently
proved that this Sp(2)/U(1) can not be positively curved in
the general sense. This is the first example to distinguish the set of compact
coset spaces admitting positively curved metrics, and that for metrics
positively curved only for commuting pairs.Comment: In this Version 2 we incorporated an argument of Burkhard Wilking,
and we modified the abstract, introduction and title to reflect that chang