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We classify the triples H ⊂ K ⊂ G of nested compact Lie groups which satisfy the "positive triple" condition that was shown in  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 2, and a family of finite quotients of T 1 S 6 .doi:10.5860/choice.35-2761a fatcat:jbwuzygphrga5bpuqkyltfc77u