Proposition 3

Let Q be a non-degenerate quadratic form of finite dimension on a vector space V over A. Let – Q denote the quadratic form on V defined by $(-Q)(x) = -Q(x)$ ($x \in V$). Then the form $Q \tau (-Q)$ is neutral.

Indeed the restriction of $Q \tau (-Q)$ to the diagonal D of $V \times V$ is zero. The index of this form is therefore $\geqslant \frac{1}{2} \dim (V \times V)$ (§ 4, no 2, déf. 2), and consequently is equal to $\frac{1}{2} \dim (V \times V)$ (ibid., formula (4)). It follows that $Q \tau (-Q)$ is neutral (ibid.).

This permits us to give the following definition: