Let \(g\) be an invertible element of a universal Clifford algebra \(A\) such that, for each \(x \in X, g x \widehat {g}^{-1} \in X\). Then the map
\[
\rho _{X, g}: x \mapsto g x \widehat {g}^{-1}
\]
is an orthogonal automorphism of \(X\).
The element \(g\) will be said to induce or represent the orthogonal transformation \(\rho _{X, g}\) and the set of all such elements \(g\) will be denoted by \(\Gamma (X)\) or simply by \(\Gamma \).
The subset \(\Gamma \) is a subgroup of \(A\).
The group \(\Gamma \) is called the Clifford group (or Lipschitz group) for \(X\) in the Clifford algebra \(A\). Since the universal algebra \(A\) is uniquely defined up to isomorphism, \(\Gamma \) is also uniquely defined up to isomorphism.
An element \(g\) of \(\Gamma (X)\) represents a rotation of \(X\) if and only if \(g\) is the product of an even number of elements of \(X\). The set of such elements will be denoted by \(\Gamma ^0=\Gamma ^0(X)\). An element \(g\) of \(\Gamma \) represents an anti-rotation of \(X\) if and only if \(g\) is the product of an odd number of elements of \(X\). The set of such elements will be denoted by \(\Gamma ^1=\Gamma ^1(X)\). Clearly, \(\Gamma ^0=\Gamma \cap A^0\) is a subgroup of \(\Gamma \), while \(\Gamma ^1=\Gamma \cap A^1\).
The Clifford group \(\Gamma (X)\) of a quadratic space \(X\) is larger than is necessary if our interest is in representing orthogonal transformations of \(X\). Use of a quadratic norm \(N\) on the Clifford algebra \(A\) leads to the definition of subgroups of \(\Gamma \) that are less redundant for this purpose. This quadratic norm \(N: A \rightarrow A\) is defined by the formula
\[
N(a)=a^{-} a, \text { for any } a \in A,
\]
For \(X\) and \(\Gamma =\Gamma (X)\) as above we now define
\[
\operatorname {Pin} X=\{g \in \Gamma :|N(g)|=1\} \text { and } \operatorname {Spin} X=\left \{g \in \Gamma ^0:|N(g)|=1\right \} \text {. }
\]