Lemma. Pin-range refinement of the correction [cartan1981theory, Section 10, pp. 10--12]; [lawson2016spin, I.2, (2.26) and Theorem 2.9, pp. 18--19] [fcap-0018]
Lemma. Pin-range refinement of the correction [cartan1981theory, Section 10, pp. 10--12]; [lawson2016spin, I.2, (2.26) and Theorem 2.9, pp. 18--19] [fcap-0018]
Under the hypotheses of Lemma [fcap-000V], suppose moreover that \(K\) is separably closed and take \(H\) to be the range of the Pin action. Over a separably closed field, TauCeti's fixed-sign square condition from Theorem [fcap-000X] holds, so every reflection factor used by the correction has a square-normalized Pin lift by Lemma [fcap-000B]. The correcting element \(r\) can therefore be chosen in \[\operatorname {range}(\operatorname {pinToOrthogonal}),\] and still satisfies \(rg\vert _{W+Kx}=\operatorname {id}\). The statement concerns the correcting element in the Pin-action range; in the exceptional case it is the image of a product of two Pin lifts, not the image of a single reflecting vector.