We consider $R \in \mathrm{O}_{d}(\mathbb{R})$ with $\operatorname{det}(R) = -1$, and the notation from question 23. We set $S_{0} = (S_{ij})_{1 \leqslant i,j \leqslant d-1} \in \mathscr{M}_{d-1}(\mathbb{R})$.
[(a)] Show that $\langle D, R \rangle = \operatorname{tr}(S_{0} R_{0}) - S_{dd}$.
[(b)] Show that $\operatorname{tr}(S_{0} R_{0}) \leqslant \operatorname{tr}(S_{0})$.
[(c)] Show that $\operatorname{tr}(S_{0}) + S_{dd} = \operatorname{tr}(D)$ and deduce that $\langle D, R \rangle \leqslant \operatorname{tr}(D) - 2S_{dd}$.
We consider $R \in \mathrm{O}_{d}(\mathbb{R})$ with $\operatorname{det}(R) = -1$, and the notation from question 23. We set $S_{0} = (S_{ij})_{1 \leqslant i,j \leqslant d-1} \in \mathscr{M}_{d-1}(\mathbb{R})$.
\begin{itemize}
\item[(a)] Show that $\langle D, R \rangle = \operatorname{tr}(S_{0} R_{0}) - S_{dd}$.
\item[(b)] Show that $\operatorname{tr}(S_{0} R_{0}) \leqslant \operatorname{tr}(S_{0})$.
\item[(c)] Show that $\operatorname{tr}(S_{0}) + S_{dd} = \operatorname{tr}(D)$ and deduce that $\langle D, R \rangle \leqslant \operatorname{tr}(D) - 2S_{dd}$.
\end{itemize}