We propose to prove Witt's theorem, whose statement is: ``let $F$ and $F'$ be two vector subspaces of $E$ such that there exists an isometry $f$ from $(F, q_{/F})$ to $(F', q_{/F'})$. Then there exists $g \in O(E,q)$ such that $g_{/F} = f$.''
We now assume that $F$ and $F'$ are non-singular, with $\operatorname{dim}(F) = \operatorname{dim}(F') > 1$.
a) Show that there exist $F_1$ and $F_2$ non-singular, such that $F_1 \perp F_2$ and $F = F_1 \oplus F_2$, with $\operatorname{dim}(F_1) = \operatorname{dim}(F) - 1$.
b) Suppose that there exists $g \in O(E,q)$ such that $g_{/F_1} = f_{/F_1}$. Denote $F_1' = f(F_1)$. Show that $f(F_2) \subset F_1'^\perp$ and that $g(F_2) \subset F_1'^\perp$.
c) Show that there exists
$$h \in O\left(F_1'^\perp, q_{/F_1'^\perp}\right) \text{ such that } h_{/g(F_2)} = (f \circ g^{-1})_{/g(F_2)}.$$
d) Show that there exists $k \in O(E,q)$ such that $k_{/F} = f$.