= 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),
$$
F^{-1}(0)=V\cap O(n).
$$
The <regular level set theorem> makes this a <submanifold> of codimension $\dim\mathcal S=n(n+1)/2$, and hence
$$
\boxed{\dim O(n)=n(n-1)/2}.
$$
This 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.
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