The universality property

Chevalley [Che] has pointed out the usefulness of the following property of Clifford algebras, which is an immediate consequence of their definition.

Here are a few applications of universality that yield several useful operations on the Clifford algebra.

1. Grading: take $A = \Cl(V, g)$ itself; the linear map $v \mapsto -v$ on $V$ extends to an automorphism $\chi \in \Aut(\Cl(V,g))$ satisfying $\chi^2 = \id_A$, given by
   
   $$\chi(v_1 \dots v_r) := (-1)^r v_1 \dots v_r.$$
   
   
   This operator gives the $\bZ_2$-grading
   
   $$\Cl(V, g) =: \Cl^0(V, g) \oplus \Cl^1(V, g).$$
   
   

2. Reversal: take $A = \Cl(V, g)^\opp$, the opposite algebra. Then the map $v \mapsto v$, considered as the inclusion $V \hookto A$, extends to an antiautomorphism $a \mapsto a^!$ of $\Cl(V, g)$, given by $(v_1 v_2 \dots v_r)^! := v_r \dots v_2 v_1$.
3. Complex conjugation: the complexification of $\Cl(V, g)$ is $\Cl(V, g) \ox_\bR \bC$, which is isomorphic to $\Cl(V^\bC, g^\bC)$ as a $\bC$-algebra. Now take $A$ to be $\Cl(V, g) \ox_\bR \bC$ and define $f\: v \mapsto \bar v : V^\bC \to V^\bC \hookto A$ (a real-linear map). It extends to an antilinear automorphism of $A$. Note that Lemma  guarantees $\bR$-linearity, but not $\bC$-linearity, of the extension even when $A$ is a $\bC$-algebra.
4. Adjoint: Also, $a^* := (\bar a)^!$ is an antilinear involution on $\Cl(V, g) \ox_\bR \bC$.

5. Charge conjugation: $\ka(a) := \chi(\bar a) : v_1 \dots v_r \mapsto (-1)^r \bar v_1 \dots \bar v_r$ is an antilinear automorphism of $\Cl(V, g) \ox_\bR \bC$.