Example. The compact three-dimensional Spin group is the unit quaternions [fcap-002H]

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Let \(Q_{3,0}(x_0,x_1,x_2)=x_0^2+x_1^2+x_2^2\) and write \(e_0,e_1,e_2\) for its coordinate basis. TauCeti's chosen algebra equivalence \(\theta :\mathcal {C}\kern -2pt\ell ^{+}(Q_{3,0})\simeq \mathbb H\) has the exact axes \[e_0e_1\longmapsto -k,\qquad e_0e_2\longmapsto i,\qquad e_1e_2\longmapsto j\] TauCeti.realCliffordThreeZeroEvenEquivQuaternion TauCeti.realCliffordThreeZeroEvenEquivQuaternion_ι_single_zero_one TauCeti.realCliffordThreeZeroEvenEquivQuaternion_ι_single_two_zero TauCeti.realCliffordThreeZeroEvenEquivQuaternion_ι_single_one_two. Hence \[\theta (s+A e_0e_1+B e_0e_2+C e_1e_2) =s+B i+C j-A k.\] Under \(\theta \), reversal is quaternion conjugation and reverse norm is \[s^2+A^2+B^2+C^2\] TauCeti.realCliffordThreeZeroEvenEquivQuaternion_reverseEven TauCeti.realCliffordThreeZeroEvenEquivQuaternion_map_reverseEven_mul_self_eq_normSq.

Let \(\mathbb H^1=\{q\in \mathbb H:\bar q q=1\}\). Restricting \(\theta \) to Spin gives a homomorphism \[\operatorname {Spin}(Q_{3,0})\longrightarrow \mathbb H^1.\] For \(q=r+x i+y j+z k\in \mathbb H^1\), the inverse Clifford value is \[r-z e_0e_1+x e_0e_2+y e_1e_2.\] Its reverse norm is one, so the dimension-three carrier lemma places it in the actual Spin group. The algebra inverse proves both compositions and multiplicativity. Therefore \[\operatorname {Spin}(Q_{3,0})\simeq \mathbb H^1.\] TauCeti separately packages the algebra map as an equivalence between the even-unitary carrier and unit quaternions TauCeti.realCliffordThreeZeroEvenUnitaryEquivQuaternionUnitary; the preceding carrier lemma is what identifies that carrier with Spin in this real dimension-three case.

In [gallier2014clifford, Section 1.4, pp. 28--29], the analogous compact calculation writes the even element, obtains its sum-of-four-squares norm, and identifies the unit quaternions. Its generators square to \(-1\). Likewise, [lawson2016spin, I.3, Theorem 3.7, pp. 23--24; I.4, (4.0), p. 25] uses \(v^2=-q(v)\) to reduce an even algebra and identify the quaternion base entry. Translating by \(Q=-q\) and the even sign-change equivalence gives the same compact algebra and norm. The displayed axis signs are the exact TauCeti choice above, rather than a relabeling imported from either source.