distribution on the Stiefel manifold, and such that the set $\mathbb G$ is right-invariant under the action of $\mathcal O_p$, i.e., $\mathbb G R = \mathbb G$ whenever $R \in \mathcal O_d$. Here, the constant $\gamma = \gamma(D)$ depends only on $D$, and the constant $\kappa = \kappa(E)$ depends only on $E$.

For any full rank $d\times p$ matrix $M$, any symmetric positive definite $d\times d$ matrix $\Sigma$ and $f_{\tilde{z}}\in\mathcal F_{d,20}(D,E)$, we define the set $$\mathbb U \;:=\; \mathbb U(M,\Sigma, f_{\tilde{z}}) \;:=\; 
\left\{ R\in \mathcal O_d : R'\Sigma^{1/2} M (M'\Sigma M)^{-1/2} 
    \in \mathbb G(f_{\tilde{z}})\right\}.$$ Now take a random matrix $U$ that is uniformly distributed on $\mathcal O_d$ and another random matrix $V$ that is uniformly distributed on $\mathcal O_p$, such that $U$ and $V$ are independent, and note that by right-invariance of $\mathbb G$, $$\begin{aligned}
\nu_d(\mathbb U) \; &= \; {{\mathbb P}}(U \Sigma^{1/2}M(M'\Sigma M)^{-1/2} \in \mathbb G)\\
\;&=\; {{\mathbb P}}(U \Sigma^{1/2}M(M'\Sigma M)^{-1/2} V \in \mathbb G)
\;=\; \nu_{d,p}(\mathbb G),\end{aligned}$$ because $\Sigma^{1/2}M(M'\Sigma M)^{-1/2}\in\mathcal V_{d,p}$ and $\nu_{d,p}$ is characterized by left and right invariance under the appropriate orthogonal groups. It follows that $\nu_d(\mathbb U^c)$ is bounded by the expression on the right-hand side of whenever $f_{\tilde{z}} \in \mathcal F_{d,20}(D,E)$, which establishes the 