Lemma. Reverse norm one is the Spin carrier in real dimension four [fcap-002J]

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Let \(V\) be a four-dimensional real vector space and let \(Q\) be nondegenerate. Suppress the canonical generator map in Clifford products. For \(x\in \mathcal {C}\kern -2pt\ell ^{+}(Q)\), the following are equivalent: \[x\text { is the Clifford value of an element of }\operatorname {Spin}(Q), \qquad \operatorname {rev}(x)x=1.\] The forward implication follows from the exact Spin-carrier description CliffordAlgebra.range_spinGroup_toUnits.

Conversely, suppose \(\operatorname {rev}(x)x=1\). Since the Clifford algebra is finite-dimensional, the displayed left inverse is also a right inverse, so \(x\) is a unit and \(x^{-1}=\operatorname {rev}(x)\). For \(v\in V\), put \(w=xvx^{-1}\). The element \(w\) is odd, and \[\operatorname {rev}(w) =\operatorname {rev}(x^{-1})v\operatorname {rev}(x) =xvx^{-1}=w.\] In an orthogonal-basis Clifford expansion in dimension four, the odd part has only degrees one and three. Reversal is \(+1\) on degree one and \(-1\) on degree three, so \(w\in V\). Gallier uses this observation in the compact calculation [gallier2014clifford, p. 30]; the same basis-and-reversal argument applies to every nondegenerate real form in dimension four. Also \(w^2=Q(v)1\). Conjugation is injective, and its restriction to the finite-dimensional space \(V\) is therefore an automorphism. Thus it defines \(g\in O(V,Q)\).

Choose \(y\) in the vector-generated Lipschitz group whose twisted action is \(g\) CliffordAlgebra.lipschitzToOrthogonal_surjective, and set \(z=y^{-1}x\). Since \(x\) is even, equality of the two actions rearranges for the grading involution \(\alpha \) to \[\alpha (z)v=vz\qquad (v\in V).\] The graded-centrality theorem therefore gives \(z=c1\) for some real \(c\) CliffordAlgebra.exists_eq_algebraMap_of_involute_mul_ι_eq_ι_mul. The unit \(z\) has \(c\ne 0\). Nondegeneracy and positive dimension supply a nonisotropic vector, so every nonzero scalar unit belongs to the generated Lipschitz group CliffordAlgebra.unitsMap_algebraMap_mem_lipschitzGroup. Thus \(x=yz\) is Lipschitz. It is even and has reverse norm one, hence lies in Spin by Remark [fcap-0028]. This dimension-sensitive argument is not asserted in dimension five or six.