Definition 1

The Clifford algebra of $Q$, denoted by $C(Q)$, is the quotient algebra of the tensor algebra $T(E)$ of the module $E$ by the two-sided ideal (denoted by $I(Q)$) generated by the elements of the form $x \otimes x - Q(x)\cdot 1$ $(x \in E)$.

We shall denote by $\rho_Q$ (or simply by $\rho$ when there is no danger of confusion) the mapping of $E$ into $C(Q)$ composed of the canonical mapping of $E$ into $T(E)$ and the canonical mapping $\sigma$ of $T(E)$ onto $C(Q)$; the mapping $\rho_Q$ is said to be canonical. Let us note that $C(Q)$ is generated by $\rho_Q(E)$, and that, for $x \in E$, we have

whence, replacing $x$ by $x + y$ ($x,\ y$ in $E$)