Consider the orthogonal group $O(n)$ as a Riemannian manifold endowed with the usual (bi-invariant) metric $\langle P, Q \rangle_A = \textrm{Tr}\ P^\top Q$ for tangent vectors $P, Q$, with $$T_A O(n) = \{ P \in \mathbb{R}^{n \times n} : A^\top P\ \text{is skew-symmetric} \}.$$

I'm interested in its sectional curvatures. For $P, Q$ orthonormal, they are given by (see, e.g., [1, Corollary 3.19])
$$K(P, Q) = \frac{\Big\lVert [P, Q] \Big\rVert_F^2}{4}.$$

From this, it's clear that $K(P, Q) \ge 0$, but **can we derive an upper bound too**? I *think* it should be less than $1/4$, but I'm not sure how to prove it.  Sorry if it's really obvious!


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[1]: Jeff Cheeger and David G Ebin. Comparison theorems in Riemannian geometry,
volume 365. American Mathematical Soc., 2008