We denote by $O(E,q)$ the set of isometries of $(E,q)$ into itself.
Let $F$ and $G$ be two vector subspaces of $E$ such that $E = F \oplus G$. We denote by $s$ the symmetry with respect to $F$ parallel to $G$.
a) Show that $s \in O(E,q)$ if and only if $F$ and $G$ are orthogonal (for $\varphi$).
b) Deduce that the symmetries in $O(E,q)$ are the symmetries with respect to $F$ parallel to $F^\perp$, where $F$ is a non-singular subspace of $E$.
c) When $H$ is a non-singular hyperplane, we will call reflection along $H$ the symmetry with respect to $H$ parallel to $H^\perp$. Show that every reflection of $E$ is an element of $O^-(E,q)$.
d) Let $(x,y) \in E^2$ such that $q(x) = q(y)$ and $q(x-y) \neq 0$. We denote by $s$ the reflection along $H = \{x-y\}^\perp$. Show that $s(x) = y$.