Let $C(Q)$ denote the clifford algebra of vector space $Q$ with respect to a quadratic form $q:V \rightarrow \Bbb R$. Hence we have the relation $w^2 = Q(w) \cdot 1$ for $w \in V$.
Let $\alpha:C(Q) \rightarrow C(Q)$ be the canonical automoprhism $\alpha^2=id, \alpha=-x$.
The Clifford group of $Q$ is $$\Gamma (Q) = \{ x \in C(Q)^* \, ; |, \alpha(x) \cdot v \cdot x^{-1} \in V \text{ for all } v \in V \}$$
How is this set closed under inverses?