The Spin group \(\operatorname {Spin}(F, \rho )\) of \(C l(F, \rho )\) is the subset of \(C l(F, \rho )\) whose elements can be written as the product \(g=u_1 \cdot \ldots \cdot u_{2 p}\) of an even number of vectors of \(F\) of norm \(\left \langle u_k, u_k\right \rangle =1\).
As a consequence : \(\langle g, g\rangle =1, g^t \cdot g=1\) and \(\operatorname {Spin}(F, \rho ) \subset O(C l)\).
The scalars \(\pm 1\) belong to the Spin group. The identity is \(+1 . \operatorname {Spin}(F, \rho )\) is a connected Lie group.