Lemma. The polarized Clifford relation [meinrenken2013clifford, Section 3.2.1, pp. 54--55] [fcap-001J]
Lemma. The polarized Clifford relation [meinrenken2013clifford, Section 3.2.1, pp. 54--55] [fcap-001J]
Creation and contraction satisfy \[\iota _y\varepsilon _x+\varepsilon _x\iota _y =B_Q(x,y)\operatorname {id}_S.\] Creation and contraction each square to zero. Parity anticommutes with both, while \(\lambda _z^2=Q(z)\operatorname {id}_S\). Consequently the operator \(c(v)\) assembled from the three polarization coordinates obeys \[c(v)^2=Q(v)\operatorname {id}_S.\] This is the quadratic form of the usual polarized anticommutation relation \(c(v)c(w)+c(w)c(v)=B_Q(v,w)\). Meinrenken states the latter as the defining condition for a Clifford module.