I am reading Kane - Reflection Groups and Invariant Theory and need help with two 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 $\displaystyle \ell $ dimensional Euclidean space ie $\displaystyle E = R^{\ell}$ where $\displaystyle R^{\ell}$ has the usual inner product (x,y).

In defining reflections with respect to vectors Kane writes:

" Given $\displaystyle 0 \ne \alpha \in E $ let $\displaystyle H_{\alpha}\subset E $ be the hyperplane

$\displaystyle H_{\alpha} = \{ x | (x, \alpha ) = 0 \} $

We then define the reflection $\displaystyle s_{\alpha} : E \longrightarrow E $ by the rules

$\displaystyle s_{\alpha} \cdot x = x $ if $\displaystyle x \in H_{\alpha} $

$\displaystyle s_{\alpha} \cdot \alpha = - \alpha $ "

Then Kane states that the following two properties follow:

(1) $\displaystyle s_{\alpha} \cdot x = x - [2 ( x, \alpha) / (\alpha, \alpha)] \alpha $ for all $\displaystyle x \in E $

(2) $\displaystyle s_{\alpha} $ is orthogonal, ie $\displaystyle ( s_{\alpha} \cdot x , s_{\alpha} \cdot y ) = (x,y) $ for all $\displaystyle x, y \in E $

I would appreciate help to show (1) and (2) above.

Peter