Example. The compact four-dimensional Spin action is left and right quaternion multiplication [fcap-003K]
Example. The compact four-dimensional Spin action is left and right quaternion multiplication [fcap-003K]
For \(Q_{4,0}=x_0^2+x_1^2+x_2^2+x_3^2\), keep the algebra map \(\Phi :\mathcal {C}\kern -2pt\ell ^+(Q_{4,0})\simeq \mathbb H\times \mathbb H\) of Example [fcap-002K]. Write \(\Phi (s)=(q_+,q_-)\) for \(s\in \operatorname {Spin}(Q_{4,0})\). In the quaternion coordinates \[W(v_0e_0+v_1e_1+v_2e_2+v_3e_3)=-v_3-v_1i+v_0j-v_2k,\] the vector action is \[W(svs^{-1})=q_+W(v)\bar q_-.\] The signed permutation \(W\) identifies the Euclidean vector norm with the quaternion norm. Both factors are unit quaternions, so the right multiplier is the inverse of \(q_-\).
To prove the formula, let \(I=e_0e_1e_2e_3\) and use the quaternion subalgebra \(H_0\) from Example [fcap-002K]. The vector \(e_3\) commutes with \(H_0\) and anticommutes with \(I\). Hence \(C(x)=e_3xe_3\) interchanges the two components of \(\Phi (x)\). The identities \[e_0e_3=-I e_1e_2,\qquad e_1e_3=I e_0e_2,\qquad e_2e_3=-I e_0e_1\] give \(\Phi _1(ve_3)=-W(v)\). Now \[(svs^{-1})e_3=s(ve_3)C(s^{-1}).\] Applying \(\Phi _1\) and negating proves the action formula. For the diagonal pair \((q,q)\), the vector \(e_3\) is fixed because \(W(e_3)=-1\); on its orthogonal complement \(W\) restricts to \(\eta _3\) of Example [fcap-003J]. Thus the three-dimensional conjugation action is the diagonal case.
TauCeti chooses a different algebra equivalence, denoted here by
\(\widehat \Phi \). Its bilinear-generator formula gives
\[\begin {aligned}
e_0e_1&\longmapsto (-k,-k),& e_0e_2&\longmapsto (j,-j),\\
e_0e_3&\longmapsto (i,i),& e_1e_2&\longmapsto (-i,i),\\
e_1e_3&\longmapsto (j,j),& e_2e_3&\longmapsto (k,-k)
\end {aligned}\]
TauCeti.realCliffordFourZeroEvenEquivQuaternionProd_ι.
In particular, \(\widehat \Phi (I)=(1,-1)\). Define quaternion automorphisms
\[\begin {aligned}
A(a+bi+cj+dk)&=a-ci+bj+dk,\\
B(a+bi+cj+dk)&=a+ci-bj+dk.
\end {aligned}\]
They are conjugation by \((1+k)/\sqrt {2}\) and \((1-k)/\sqrt {2}\),
respectively, and satisfy \(B(q)=kA(q)k^{-1}\). The rows for
\(e_0e_1,e_0e_2,e_1e_2\) identify the restrictions to \(H_0\) as
\(A\circ \theta _0\) and \(B\circ \theta _0\). Together with the image of
\(I\), the decomposition \(x=u+Iu'\) therefore proves
\[\widehat \Phi (x)=\bigl (A(\Phi _1(x)),B(\Phi _2(x))\bigr ).\]
TauCeti's vector coordinates are
\[\eta _4(v)=-v_2+v_1i-v_0j+v_3k=A(W(v))k^{-1}\]
TauCeti.realCliffordFourZeroQuaternionEquiv_apply.
Since \(A,B\) commute with quaternion conjugation and
\(A(\bar q)k^{-1}=k^{-1}\overline {B(q)}\), the formula for \(W\) gives
\[\eta _4(svs^{-1})=A(q_+)\eta _4(v)\overline {B(q_-)}.\]
This is precisely the action in the \(\widehat \Phi \) coordinates
TauCeti.realSpinFourEquivQuaternionUnitaryProd_action.
The displayed changes of coordinates derive its relation to the natural
pair \(\Phi \) used in Example [fcap-002K].
Gallier [gallier2014clifford, Section 1.4, pp. 29--30] gives the compact even-algebra decomposition into two quaternion factors with negative-square generators; Example [fcap-002K] uses the positive-square convention. Porteous [porteous1995clifford, Chapter 8, Propositions 8.26--8.27, p. 65] describes the four-dimensional rotations \(z\mapsto qz\bar r\) for unit quaternions \(q,r\). The calculation with \(ve_3\) identifies these left and inverse-right factors for \(\Phi \); the explicit \(A,B,W,\eta _4\) formulas then give the coordinate translation to \(\widehat \Phi \).