Proposition 3

Let Q and Q' be two quadratic forms and F a bilinear form on E such that $Q'(x) = Q(x) + F(x, x)$ for every $x \in E$. The mapping $\lambda_F$ maps the ideal $I(Q')$ onto the ideal $I(Q)$, and defines an isomorphism (denoted by $\overline{\lambda_F}$) of the A-module $C(Q')$ onto the A-module $C(Q)$.

Since $\lambda_F$ is a bijection whose inverse bijection is $\lambda_{-F}$ (Lemma 3), it suffices to prove the inclusion $\lambda_F(I(Q')) \subset I(Q)$. Since $I(Q)$ is a left ideal stable under $i_x^F$ (Lemma 1), (8) shows that the set of $u \in T(E)$ such that $\lambda_F(u) \in I(Q)$ is a left ideal. It therefore suffices to prove that, whatever $u \in T(E)$ and $x \in E$, we have $\lambda_F(x \otimes x \otimes u - Q'(x)u) \in I(Q)$. Now, according to (8) and Lemma 1, we have

whence