Proposition 1

*Let f be a linear mapping of E into an algebra D over A such that $f(x)^2 = Q(x).1$ for all $x \in E$. There exists one and only one homomorphism $\bar{f}$ of $C(Q)$ into D such that $f = \bar{f} \circ \rho_Q$.

The uniqueness of $\bar{f}$ follows from the fact that $C(Q)$ is generated by $\rho_Q(E)$. Let $h$ be the unique homomorphism of $T(E)$ into D which extends $f$ ($h$ is defined by $h(x_1 \otimes \cdots \otimes x_n) = f(x_1) \ldots f(x_n)$). We have

and consequently $h$ vanishes on $I(Q)$ and defines by passing to the quotient the homomorphism $\bar{f}$ in question.

Prop. 1 expresses that $C(Q)$ is the solution of a universal mapping problem (Sets, Chap. IV, § 3, No. 1).

Take in particular for D the opposite algebra of $C(Q)$ and for $f$ the mapping $\rho$; prop. 1 entails that there exists one and only one anti-automorphism $\beta$ of $C(Q)$ whose restriction to $\rho(E)$ is the identity; it is called the principal antiautomorphism of $C(Q)$. It is clear that $\beta^2 = 1$.

D’autre part let $Q'$ be a quadratic form on an A-module $E'$ and $f$ a linear mapping of $E$ into $E'$ such that $Q' \circ f = Q$. We have $\rho_Q(f(x))^2 = Q'(f(x)).1 = Q(x).1$, and consequently there exists one and only one homomorphism $C(f)$ of $C(Q)$ into $C(Q')$ such that $C(f) \circ \rho_Q = \rho_{Q'} \circ f$. If $f$ is the identity, $C(f)$ is the identity; if $Q''$ is a quadratic form on an A-module $E''$ and $g$ a linear mapping of $E'$ into $E''$ such that $Q'' \circ g = Q'$, we have $C(g \circ f) = C(g) \circ C(f)$. When $E'$ is a submodule of $E$ and $f$ the canonical injection of $E'$ into $E$ (so that $Q'$ is the restriction of $Q$ to $E'$), we say that $C(f)$ is the canonical homomorphism of $C(Q')$ into $C(Q)$.

Let us take in particular $Q' = Q$ and for $f$ the mapping $x \to -x$; we see that there exists one and only one automorphism $\alpha$ of $C(Q)$ such that $\alpha \circ \rho = -\rho$; it is called the principal automorphism of $C(Q)$. It is clear that $\alpha^2 = 1$, and that the restriction of $\alpha$ to $C^+$ (resp. $C^-$) is the identity (resp. the mapping $u \to -u$).