definition. Lipschitz-Clifford group [pr-spin] [spin-000W]

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The Lipschitz-Clifford group is defined as the subgroup closure of all the invertible elements in the form of \(\iota _Q(m)\), \[ \Gamma \equiv \left \{ x_1 \ldots x_k \in \mathcal {C}\kern -2pt\ell ^{\times }(Q) \mid x_i \in V \right \} \] where \[ \mathcal {C}\kern -2pt\ell ^{\times }(Q) \equiv \left \{ x \in \mathcal {C}\kern -2pt\ell (Q) \mid \exists x^{-1} \in \mathcal {C}\kern -2pt\ell (Q), x^{-1} x = x x^{-1}=1\right \} \] is the group of units (i.e. invertible elements) of \(\mathcal {C}\kern -2pt\ell (Q)\), \[ V \equiv \left \{ \iota _Q(m) \in \mathcal {C}\kern -2pt\ell (Q) \mid m \in M \right \} \] is the subset \(V\) of \(\mathcal {C}\kern -2pt\ell (Q)\) in the form of \(\iota _Q(m)\).