Example. The compact three-dimensional Spin action is quaternion conjugation [fcap-003J]

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For \(Q_{3,0}=x_0^2+x_1^2+x_2^2\), use the algebra map \(\theta :\mathcal {C}\kern -2pt\ell ^+(Q_{3,0})\simeq \mathbb H\) of Example [fcap-002H]. The vector isometry to pure quaternions is \[\eta _3(v_0e_0+v_1e_1+v_2e_2)=-v_1i+v_0j-v_2k.\] For \(s\in \operatorname {Spin}(Q_{3,0})\) and \(q=\theta (s)\in \mathbb H^1\), the ordinary Clifford conjugation action satisfies \[\eta _3(svs^{-1})=q\eta _3(v)\bar q.\] These are the coordinates and action in TauCeti TauCeti.coe_realCliffordThreeZeroPureQuaternionEquiv_apply TauCeti.realSpinThreeEquivQuaternionUnitary_action.

The volume element \(\omega =e_0e_1e_2\) is central and has \(\omega ^2=-1\). Multiplying a vector by it gives \[v\omega =v_0e_1e_2+v_1e_2e_0+v_2e_0e_1,\qquad \eta _3(v)=\theta (v\omega ).\] The three coefficients are only permuted and signed, so \(\eta _3\) is an isometry onto the pure quaternions. Since \(\omega \) is central, \[(svs^{-1})\omega =s(v\omega )s^{-1}.\] Apply \(\theta \); reversal on the even algebra becomes quaternion conjugation, and \(q^{-1}=\bar q\). This proves the displayed action formula.

The axes also determine the sign of the rotation. For real \(t\), put \(s=\cos t+\sin t\,e_0e_1\). Its reverse norm is one, hence it belongs to Spin by Example [fcap-002H], and \(q=\cos t-\sin t\,k\). Direct multiplication gives \[\begin {aligned} se_0s^{-1}&=\cos (2t)e_0-\sin (2t)e_1,\\ se_1s^{-1}&=\sin (2t)e_0+\cos (2t)e_1,\\ se_2s^{-1}&=e_2. \end {aligned}\] In quaternion coordinates the first identity is \(qj\bar q=\cos (2t)j+\sin (2t)i\), since \(\eta _3(e_1)=-i\). Thus the rotor parameter is a half-angle with the orientation fixed by the stated Clifford and quaternion axes.

Gallier [gallier2014clifford, Section 1.4, p. 29] proves the same volume-element transport to quaternion conjugation with compact generators of square \(-1\). His volume has square \(+1\), and his map is \(v\mapsto -\omega v\); for the positive squares here it is \(v\mapsto v\omega \), followed by the axes of Example [fcap-002H]. Porteous [porteous1995clifford, Chapter 8, Propositions 8.19--8.21, pp. 62--63] describes the resulting unit-quaternion rotations of pure quaternions. The signed half-angle calculation above specifies this action in the coordinates used here.