  A compact Riemannian homogeneous space $G/H$, with a bi--invariant orthogonal
decomposition $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ is called positively
curved for commuting pairs, if the sectional curvature vanishes for any tangent
plane in $T_{eH}(G/H)$ spanned by a linearly independent commuting pair in
$\mathfrak{m}$. In this paper,we will prove that on the coset space
$\mathrm{Sp}(2)/\mathrm{U}(1)$, in which $\mathrm{U}(1)$ corresponds to a short
root, admits positively curved metrics for commuting pairs. B. Wilking recently
proved that this $\mathrm{Sp}(2)/\mathrm{U}(1)$ 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.
