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) $$ \operatorname{sec}_{\operatorname{Gr}(n,k)}(X, Y) \leq 4. $$

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.

On the other hand, in this paper 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?

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 of the Grassmannian with 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) $$ \operatorname{sec}_{\operatorname{Gr}(n,k)}(X, Y) \leq 4. $$

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.

On the other hand, in this paper 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?