Recall that \(\Pi : \mathrm {Cl}(V) \rightarrow \) \(\mathrm {Cl}(V), x \mapsto (-1)^{|x|} x\) denotes the parity automorphism of the Clifford algebra. Let \(\mathrm {Cl}(V)^{\times }\)be the group of invertible elements in \(\mathrm {Cl}(V)\).
The Clifford group \(\Gamma (V)\) is the subgroup of \(\mathrm {Cl}(V)^{\times }\), consisting of all \(x \in \mathrm {Cl}(V)^{\times }\) such that \(A_x(v):=\Pi (x) v x^{-1} \in V\) for all \(v \in V \subset \mathrm {Cl}(V)\).
Hence, by definition the Clifford group comes with a natural representation, \(\Gamma (V) \rightarrow \mathrm {GL}(V), x \mapsto A_x\). Let \(S \Gamma (V)=\Gamma (V) \cap \mathrm {Cl}^{\overline {0}}(V)^{\times }\)denote the special Clifford group.
The canonical representation of the Clifford group takes values in \(\mathrm {O}(V)\), and defines an exact sequence,
\[
1 \longrightarrow \mathbb {K}^{\times } \longrightarrow \Gamma (V) \longrightarrow \mathrm {O}(V) \longrightarrow 1 .
\]
It restricts to a similar exact sequence for the special Clifford group,
\[
1 \longrightarrow \mathbb {K}^{\times } \longrightarrow S \Gamma (V) \longrightarrow \mathrm {SO}(V) \longrightarrow 1 .
\]
The elements of \(\Gamma (V)\) are all products \(x=v_1 \cdots v_k\) where \(v_1, \ldots , v_k \in V\) are non-isotropic. \(S \Gamma (V)\) consists of similar products, with \(k\) even. The corresponding element \(A_x\) is a product of reflections:
\[
A_{v_1 \cdots v_k}=R_{v_1} \cdots R_{v_k} .
\]
Suppose \(\mathbb {K}=\mathbb {R}\). The Pin group \(\operatorname {Pin}(V)\) is the preimage of \(\{1,-1\}\) under the norm homomorphism \(N: \Gamma (V) \rightarrow \mathbb {K}^{\times }\). Its intersection with \(S \Gamma (V)\) is called the Spin group, and is denoted \(\operatorname {Spin}(V)\).
Since \(N(\lambda )=\lambda ^2\) for \(\lambda \in \mathbb {K}^{\times }\), the only scalars in \(\operatorname {Pin}(V)\) are \(\pm 1\). Hence, the exact sequence for the Clifford group restricts to an exact sequence,
\[
1 \longrightarrow \mathbb {Z}_2 \longrightarrow \operatorname {Pin}(V) \longrightarrow \mathrm {O}(V) \longrightarrow 1,
\]
so that \(\operatorname {Pin}(V)\) is a double cover of \(\mathrm {O}(V)\). Similarly,
\[
1 \longrightarrow \mathbb {Z}_2 \longrightarrow \operatorname {Spin}(V) \longrightarrow \mathrm {SO}(V) \longrightarrow 1,
\]
defines a double cover of \(\mathrm {SO}(V)\). Elements in \(\operatorname {Pin}(V)\) are products \(x=\) \(v_1 \cdots v_k\) with \(B\left (v_i, v_i\right )= \pm 1\). The group \(\operatorname {Spin}(V)\) consists of similar products, with \(k\) even.