transformation is an isometry. Also, and so leaves vectors parallel to invariant. So, by decomposing as a vector parallel to the vector part of and a vector normal to the vector part of and showing that the application of to the normal component of rotates it, the claim is shown. So let be the component of orthogonal to the vector part of and let . It turns out that the vector part of is given by
:.The conjugation of by can be expressed with fewer arithmetic operations as:
:

A geometric fact independent of