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
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