 Solution

The linear map $\pi (A)=(A+A^T)/2$ is the projection onto the space $\mathcal{S}$ of symmetric matrices, with kernel $\mathcal{A}$. Define $F=\pi \circ L$ on $V$. Since $DL$ is invertible everywhere on $V$ and $\pi$ is onto, $DF$ is onto everywhere. By part (ii),The regular level set theorem makes this a submanifold of codimension $\dim \mathcal{S}=n(n+1)/2$, and henceThis neighborhood is $V$ around $I$; the source's final occurrence of $U$ in this step is a neighborhood-label slip. For any $Q\in O(n)$, left multiplication by $Q$ is an invertible linear map of the ambient matrix space and carries $V\cap O(n)$ onto $QV\cap O(n)$. These translated charts establish the submanifold property at every group element.