Exercise 4.1.4

${\mathrm{\Omega }}_4=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill -i\hfill & \hfill -1\hfill & \hfill i\hfill \\ \hfill 1\hfill & \hfill -1\hfill & \hfill 1\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill i\hfill & \hfill -1\hfill & \hfill -i\hfill \end{array}\right]$

So we have $c={\mathrm{\Omega }}_{4}f=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill -i\hfill \\ \hfill -1\hfill \\ \hfill i\hfill \end{array}\right]$

Now the $F_4=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill i\hfill & \hfill -1\hfill & \hfill -i\hfill \\ \hfill 1\hfill & \hfill -1\hfill & \hfill 1\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill -i\hfill & \hfill -1\hfill & \hfill i\hfill \end{array}\right]$ we have $F_{4}c=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 4\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \end{array}\right]=4f$