I am reading Kane - Reflection Groups and Invariant Theory and need help with property A-4 of the properties of reflections stated on page 7

(see attachment - Kane _ Reflection Groups and Invariant Theory - pages 6-7)

On page 6 Kane mentions he is working in dimensional Euclidean space ie where has the usual inner product (x,y).

In defining reflections with respect to vectors Kane writes:

" Given let be the hyperplane

We then define the reflection by the rules

if

"

Then Kane states that the following property follows:

(A-4) If is an orthogonal automorphism of E then

Can someone explicitly and formally demonstrate A-4?

I am not really sure of the meaning of but I assume

And I assume that

Can anyone help me formulate an explicit and detailed proof of A-4

Peter

Peter