Show that the \(4 \times 4\) matrices \(e_{0}=I_{4}\),

\[
e_{1}=\left(\begin{array}{rrrr}
0 & -1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 \\
0 & 0 & 1 & 0
\end{array}ight), \quad e_{2}=\left(\begin{array}{rrrr}
0 & 0 & -1 & 0 \\
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0
\end{array}ight), \quad e_{3}=\left(\begin{array}{rrrr}
0 & 0 & 0 & -1 \\
0 & 0 & -1 & 0 \\
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0
\end{array}ight)
\]

satisfy Hamilton's multiplication formulae, i.e., \(e_{1} e_{2}=e_{3}, e_{2} e_{3}=e_{1}\) and so on. Hence deduce that the quaternion-to-matrix mapping

\[
q \mapsto \chi_{4}(q)=\left(\begin{array}{rrrr}
q_{0} & -q_{1} & -q_{2} & -q_{3} \\
q_{1} & q_{0} & -q_{3} & q_{2} \\
q_{2} & q_{3} & q_{0} & -q_{1} \\
q_{3} & -q_{2} & q_{1} & q_{0}
\end{array}ight)
\]

is a linear representation satisfying \(\chi_{4}(p q)=\chi_{4}(p) \chi_{4}(q)\) and \(\chi_{4}(\bar{p})=\chi_{4}^{\prime}(p)\).