 Solution

The matrix exponential is smooth, and part (a) identifies its derivative at zero as the identity. The inverse function theorem therefore gives a local diffeomorphism from an open neighborhood $U_0$ of zero onto a neighborhood of $I$. Choose a sufficiently small operator norm ball $U\subset U_0$ and let $V=\exp U$. This is again a diffeomorphism onto an open set, since it is a restriction of the original one.

The ball $U$ is preserved by $A\mapsto A^T$ and $A\mapsto -A$. Since $(\exp A{)}^T=\exp (A^T)$ and $(\exp A{)}^{-1}=\exp (-A)$, $V$ is preserved by adjoint and inversion. For its local inverse $L$, injectivity also gives