Theorem. The signature-switch recurrence
[lawson2016spin, I.4, Theorem 4.1 and (4.1), pp. 25--26] [fcap-001X]
Theorem. The signature-switch recurrence [lawson2016spin, I.4, Theorem 4.1 and (4.1), pp. 25--26] [fcap-001X]
With the TauCeti signature convention of Convention [fcap-0011], there is an algebra equivalence \[\mathcal {C}\kern -2pt\ell _{p+2,q}\simeq _{\mathbb R\text {-alg}} \mathcal {C}\kern -2pt\ell _{q,p}\otimes _{\mathbb R}M_2(\mathbb R).\] The construction first separates the last positive coordinate from \(Q_{p+2,q}\). Adjoining that positive line permits a Clifford sign switch; negating \(Q_{p+1,q}\) exchanges its positive and negative coordinate blocks. The resulting form is \(Q_{q+1,p+1}\), to which the hyperbolic recurrence of Theorem [fcap-0013] applies.
The accompanying generator theorem
SignatureSwitchRecurrenceEquiv_ι
records this composition through the positive-coordinate splitter, the
sign-switch isometry, and the generator formula for the hyperbolic
equivalence. It fixes the equivalence on the canonical Clifford generators
rather than asserting only that some algebra isomorphism exists.
After converting between the two signature conventions, inverting Lawson--Michelsohn's equation (4.1), using \(\mathcal {C}\kern -2pt\ell ^{\mathrm {Lawson}}_{0,2}\cong M_2(\mathbb R)\), and swapping the indices yields the displayed algebra isomorphism. Their equation (4.3) is instead the mixed \((1,1)\) recurrence used in Theorem [fcap-0013]. Chevalley's split-matrix and orthogonal-sum constructions give the same structural ingredients [chevalley1954algebraic, II.2.1 and II.2.5, pp. 42--46]. The exact splitter composition and generator formula are additional data recorded by TauCeti.