definition. Spin group [nlab2023spin] [spin-000Q]

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The Pin group \(\operatorname {Pin}(V ; q)\) of a quadratic vector space, is the subgroup of the group of units in the Clifford algebra \(\mathrm {Cl}(V, q)\) \[ \operatorname {Pin}(V, q) \hookrightarrow \operatorname {GL}_1(\mathrm {Cl}(V, q)) \] on those elements which are multiples \(v_1 \cdots v_n\) of elements \(v_i \in V\) with \(q\left (v_i\right )=1\).

The Spin group \(\operatorname {Spin}(V, q)\) is the further subgroup of \(\operatorname {Pin}(V ; q)\) on those elements which are even number multiples \(v_1 \cdots v_{2 k}\) of elements \(v_i \in V\) with \(q\left (v_i\right )=1\).