Let \(x=\left(x_{0}, x_{1}, x_{2}, x_{3}ight)\) be a unit vector in \(\mathbb{R}^{4}\), let \(v=\left(x_{1}, x_{2}, x_{3}ight)\) and let \(\chi(v)\) be the \(3 \times 3\) matrix in (16.25). Show that image of the mapping \(x \mapsto R(x)\)

\[
R(x)=I_{3}+2 x_{0} \chi(v)+2 \chi^{2}(v)
\]

is equal to \(\mathcal{O}_{3}^{+}\). Find the unit vector \(x^{\prime}\) such that \(R\left(x^{\prime}ight)=R^{\prime}(x)\).