For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
We assume that $h^{-1} \circ h'$ has $n$ distinct eigenvalues. Show that there exists a basis of $E$ orthogonal for both $q$ and $q'$.

Let $V$ be a $\mathbb { K }$-vector space of finite non-zero dimension. Let $I$ be a non-empty set and let $\left\{ f _ { i } \mid i \in I \right\}$ be a family of diagonalizable endomorphisms of $V$ commuting pairwise. Show that there exists a basis of $V$ in which the matrices of the endomorphisms $f _ { i }$, for $i \in I$, are diagonal. Hint: one may first treat the case where all the endomorphisms $f _ { i }$ are homotheties, then reason by induction on the dimension of $V$.