A note on quasi-positive curvature conditions

We classify the triples $H \subset K \subset G$ of nested compact Lie groups which satisfy the "positive triple" condition that was shown by the second author to ensure that $G/H$ admits a metric with quasi-positive curvature. A few new examples of spaces that admit quasi-positively curved metrics emerge from this classification; namely, a $\CP^2$-bundle over $S^6$, a $B^7$-bundle over $\HP^2$, a $\CP^{2n-1}$-bundle over $\HP^{n}$ for each $n\geq 2$, and a family of finite quotients of $T^1S^6$.