definition. Spin group [wiki2023spin] [spin-000P]

✍️source

The pin group \(\operatorname {Pin}(V)\) is a subgroup of \(\mathrm {Cl}(V)\) 's Clifford group of all elements of the form \[ v_1 v_2 \cdots v_k \] where each \(v_i \in V\) is of unit length: \(q\left (v_i\right )=1\).

The spin group is then defined as \[ \operatorname {Spin}(V)=\operatorname {Pin}(V) \cap \mathrm {Cl}^{\text {even }}, \] where \(\mathrm {Cl}^{\text {even }}=\mathrm {Cl}^0 \oplus \mathrm {Cl}^2 \oplus \mathrm {Cl}^4 \oplus \cdots \) is the subspace generated by elements that are the product of an even number of vectors. That is, \(\operatorname {Spin}(V)\) consists of all elements of \(\operatorname {Pin}(V)\), given above, with the restriction to \(k\) being an even number. The restriction to the even subspace is key to the formation of two-component (Weyl) spinors, constructed below.