Proving the Orthogonality of linear transformation

Question

I am having trouble with the second part of the following problem.

This is what I have tried.

How can I show that the map is norm preserving. This last vector is now hard to deal with. What could be the possible mistake?

Any suggestions will be really helpful.

Thanks & regards

Answer 1

If $z=\bigl(a,(b,c,d)\bigr)$, with $a^2+b^2+c^2+d^2=1$, and $v=\bigl(x,(y,z,t)\bigr)$, then$$zv=\bigl(a x-b y-c z-d t,(a y+b x+c t-d z,a z-b t+c x+d y,a t+b z-c y+dx)\bigr)$$and therefore\begin{align}\|zv\|^2&=(a^2+b^2+c^2+d^2)(x^2+y^2+z^2+t^2)\\&=x^2+y^2+z^2+t^2\\&=\|v\|^2.\end{align}So,$$\|zv\overline z\|^2=\left\|z\left(v\overline z\right)\right\|=\left\|v\overline z\right\|^2=\|v\|^2.$$