The "group of units" in \(C l_{p, q}\), denoted \(C l_{p, q}^{\times }\), is the group of all invertible elements.
An important subgroup of \(C l^{\times }(V, q)\) is the group \(P(V, q)\) generated by elements \(v \in V\) with \(q(v) \neq 0\). Quotienting out by constants, we obtain the Pin group. Specifically, \(\operatorname {Pin}(V, q)\) (or \(\operatorname {Pin}_{p, q}\) ) is the group generated by elements \(v \in V\) with \(q(v)= \pm 1\). Further define the spin groups to be
\[
\operatorname {Spin}(V, q)=\operatorname {Pin}(V, q) \cap C l^0(V, q) .
\]