We show that a left invariant metric on a compact Lie group $G$ which is obtained by stretching a biinvariant metric in the direction of a subalgebra $\h$ of $\g$ always has some negative sectional curvature, unless the semi-simple part of $\h$ is an ideal of $\g$.
Riemannian submersions from simple, compact Lie groups
Sobolev Maps into Compact Lie Groups and Curvature
Average signature of geodesic paths in compact Lie groups
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