The following exercise problem is contained in O'neill's text.
(a) is directly derived from the computations with the definition of the isometry.
However, there is something wrong in (b).
To determine $F$, we need to get $T$,$C$. ($T$:translation part, $C$:orthogonal part).
As $C$ depends on the choice of frame(the sort of basis), $T$ is not determined uniquely.
I think this is not good problem. Am I right?