Consider the real Grassmannian as the symmetric space $\operatorname{Gr}(n,k) \cong \operatorname{O}(n)/(\operatorname{O}(k) \times \operatorname{O}(n-k))$ for $n \geq 3$, $k \geq 2$, where the metric is that induced from the bi-invariant metric on $\operatorname{O}(n)$, $\langle X,Y\rangle =\frac{1}{2}\operatorname{tr}(X^\intercal Y)$. The sectional curvature on $\operatorname{O}(n)$ with this bi-invariant metric is given by

$$
\operatorname{sec}_{\operatorname{O}(n)}(X, Y) = \frac{1}{4}\lVert[X,Y]\rVert^2.
$$
where the norm is that induced by the scalar product.

Writing $\mathfrak{h} = \mathfrak{o}(n-k) \oplus \mathfrak{o}(k)$ and $\mathfrak{m} = \mathfrak{h}^\perp \subset \mathfrak{o}(n)$, by the O'Neill's formula and identifying a tangent space to the Grassmannian to a subspace of the Lie algebra of $\operatorname{O}(n)$, we have that the sectional curvature of $\operatorname{Gr}(n,k)$ for a pair of orthonormal vectors $X, Y \in \mathfrak{m}$

$$
\operatorname{sec}_{\operatorname{Gr}(n,k)}(X, Y) = \frac{1}{4}\lVert[X,Y]\rVert^2 + \frac{3}{4}\lVert [X,Y]_{\mathfrak{h}}\rVert^2 = \lVert[X,Y]\rVert^2
$$
since $[\mathfrak{m}, \mathfrak{m}] \subset \mathfrak{h}$.
Using now the bounds for the Lie bracket in $\operatorname{O}(n)$ (see [this other MO answer][1])
$$
\operatorname{sec}_{\operatorname{Gr}(n,k)}(X, Y) \leq 4.
$$

This bound is not tight, as it can be seen by looking at the equality cases for the inequality used, as per Lemma 2.5 in [this article][2].

On the other hand, in [this paper][3] the author announces (without proof) in Theorem 3a that
$$
\operatorname{sec}_{\operatorname{Gr}(n,k)}(X, Y) \leq 2.
$$
and even gives examples in Theorem 5a of a submanifold where this bound is achieved.

Is there any reference in which the tighter bound of $2$ is computed? Is there a reference where the tightness of the bound $2$ is also derived?


  [1]: https://mathoverflow.net/a/343406/145929
  [2]: https://www.sciencedirect.com/science/article/pii/S000187081300385X
  [3]: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC539132/pdf/pnas00119-0088.pdf