 Solution

Part (b) already shows $\exp (U\cap \mathcal{A})\subset V\cap O(n)$. Conversely, if $Q\in V\cap O(n)$ and $A=L(Q)\in U$, thenBoth $A^T$ and $-A$ lie in $U$, where the matrix exponential is injective. Therefore $A^T=-A$, proving the reverse inclusion. The restricted map is a bijection, indeed a diffeomorphism between these local sets.