We consider $R \in \mathrm{O}_{d}(\mathbb{R})$ with $\operatorname{det}(R) = -1$.
\begin{itemize}
\item[(a)] Show that there exists an orthonormal basis $(u_{1}, \ldots, u_{d})$ of $\mathbb{R}^{d}$ such that $Ru_{d} = -u_{d}$ and $u_{d}^{T} Rx = 0$ for all $x \in E_{1}$ where $E_{1} = \operatorname{Vect}(u_{1}, \ldots, u_{d-1})$.
\item[(b)] Deduce that $R(E_{1}) \subset E_{1}$ and then that $R(E_{1}) = E_{1}$.
\end{itemize}