Convention. Real signature and Clifford signs [chevalley1954algebraic, II.2.9, pp. 65--66] [fcap-0011]
Convention. Real signature and Clifford signs [chevalley1954algebraic, II.2.9, pp. 65--66] [fcap-0011]
For \(p,q\ge 0\), put \[Q_{p,q}(x)=\sum _{i<p}x_i^2-\sum _{p\le i<p+q}x_i^2\] on \(\mathbb {R}^{p+q}\), and write \(\mathcal {C}\kern -2pt\ell _{p,q}\) for \(\mathcal {C}\kern -2pt\ell (Q_{p,q})\). Thus the first \(p\) Clifford generators square to \(+1\), and the last \(q\) square to \(-1\). This agrees with Chevalley's convention for the quadratic form and with his positive/negative inertia indices.
Lawson and Michelsohn use the same signature \(q_{r,s}\) but impose \(v^2=-q_{r,s}(v)1\) [lawson2016spin, I.3, Proposition 3.1, p. 21]. Therefore the algebras are related by the exact index swap \[\mathcal {C}\kern -2pt\ell _{p,q}=\mathcal {C}\kern -2pt\ell ^{\mathrm {Lawson}}_{q,p}.\] The recurrence below adds one index of each sign, so it is invariant under this swap.