Remarks

1) Every non-degenerate quadratic form $Q$ of finite dimension whose type is zero (i.e. such that $\theta(Q) = T_0$ with the above notation) is a neutral form. There indeed exist neutral forms $N, N'$ such that $Q \tau N$ is equivalent to $N'$. This shows that $Q$ is of even dimension, and hence that there exists a neutral form $N_1$ having the same dimension as $Q$. Since $Q$ and $N_1$ have the same type, it follows from prop. 1 that they are equivalent, and hence that $Q$ is neutral.

2) For every finite-dimensional quadratic form $Q$ over $A$, let $\delta(Q)$ denote the class modulo 2 of the dimension of $Q$. We have

Since every neutral form $N$ is of even dimension, we have $\delta(N) = 0$; the relation $Q \sim Q'$ therefore entails $\delta(Q) = \delta(Q')$. Thus the restriction of $\delta$ to the group $\mathfrak{W}$ of types of quadratic forms over $A$ is a homomorphism of $\mathfrak{W}$ into the group $\mathbf{Z}/(2)$. This homomorphism is surjective when $A$ is of characteristic $\neq 2$, but is not so if $A$ is of characteristic 2, since a quadratic form of odd dimension is then degenerate, its associated bilinear form being alternating (cf. § 5).

3) Let $a$ be an element $\neq 0$ of $A$. If $N$ is a neutral form, then so is $aN$. It follows that the relation $Q \sim Q'$ entails $aQ \sim aQ'$. For every element $T$ of the group $\mathfrak{W}$, we shall put

We thus obtain an external composition law between the group $A^*$ of the non-zero elements of $A$ and the group $\mathfrak{W}$. The following formulas result immediately from the definition:

On the other hand, if $a, b$ and $a + b$ are in $A^*$, one does not in general have $(a + b).T = a.T + b.T$ ($T \in \mathfrak{W}$).