Example. The Lorentzian four-dimensional Spin group is \(SL_2(\mathbb C)\) [fcap-002L]
Example. The Lorentzian four-dimensional Spin group is \(SL_2(\mathbb C)\) [fcap-002L]
For \(Q_{1,3}=x_0^2-x_1^2-x_2^2-x_3^2\), put \(A_j=e_0e_j\) for \(1\le j\le 3\) and \(I=e_0e_1e_2e_3\). Then the \(A_j\) square to \(1\) and anticommute, while \[I^2=-1,\qquad A_1A_2A_3=-I.\] For the Pauli matrices \[\sigma _1=\begin {pmatrix}0&1\\1&0\end {pmatrix},\quad \sigma _2=\begin {pmatrix}0&-i\\i&0\end {pmatrix},\quad \sigma _3=\begin {pmatrix}1&0\\0&-1\end {pmatrix},\] the assignments \(\Psi (A_j)=\sigma _j\) define a real-algebra map from the even Clifford algebra to \(M_2(\mathbb C)\). Since \(\sigma _1\sigma _2\sigma _3=i1\), necessarily \(\Psi (I)=-i1\). The images of \(1,A_1,A_2,A_3,I,IA_1,IA_2,IA_3\) form a real basis, so \(\Psi \) is an algebra equivalence.
Write a matrix uniquely as \[M=z_0 1+z_1\sigma _1+z_2\sigma _2+z_3\sigma _3, \qquad z_j=a_j+i b_j,\] where \[\begin {aligned} z_0&=(m_{11}+m_{22})/2,&z_3&=(m_{11}-m_{22})/2,\\ z_1&=(m_{12}+m_{21})/2,&z_2&=(m_{21}-m_{12})/(2i). \end {aligned}\] Then an explicit inverse is \[\Psi ^{-1}(M) =a_0+\sum _{j=1}^3a_jA_j -I\left (b_0+\sum _{j=1}^3b_jA_j\right ).\] Reversal fixes \(I\) and negates every \(A_j\); under \(\Psi \) it is the complex-linear matrix adjugate. Hence \[\Psi (\operatorname {rev}(x))=\operatorname {adj}(\Psi (x)),\qquad \Psi (\operatorname {rev}(x)x)=\det (\Psi (x))1.\] For an arbitrary even \(x\), this norm is algebra-valued: the determinant may be complex, corresponding to the even center. It is not yet the real scalar-valued Lipschitz norm, and adjugation is not conjugate transpose.
Restriction therefore gives a homomorphism \(\operatorname {Spin}(Q_{1,3})\to SL_2(\mathbb C)\). Conversely, a determinant-one matrix has adjugate as a two-sided inverse. Its displayed Clifford preimage is an even unit with \(\operatorname {rev}(x)x=1\), so the dimension-four carrier lemma places it in Spin. The written inverse proves both compositions and multiplicativity: \[\operatorname {Spin}(Q_{1,3})\simeq SL_2(\mathbb C).\]
The source indices require translation. [lawson2016spin, I.3, Theorem 3.7, pp. 23--24] use \(v^2=-q(v)\) and give the even-algebra reduction \[\operatorname {Cl}^{0}_{3,1}\simeq \operatorname {Cl}_{2,1}.\] Their base entries and recurrence then give \[\operatorname {Cl}_{2,1}\simeq M_2(\mathbb C)\] [lawson2016spin, I.4, (4.0) and Theorem 4.1, p. 25].
Gallier uses the opposite indexing. The positive-square \(A_j\) exhibit \(\operatorname {Cl}_{3,0}\). [gallier2014clifford, Example 1.3, p. 33] records the matrix base entries, while [gallier2014clifford, Lemma 1.15, p. 34] supplies the recurrence. Together they give \[\operatorname {Cl}_{3,0} \simeq \operatorname {Cl}_{0,1}\otimes \operatorname {Cl}_{2,0} \simeq M_2(\mathbb C).\]
[porteous1995clifford, Chapter 17, Proposition 17.5, pp. 151--152] identifies his norm-one subgroup \(\operatorname {Spin}^{+}(1,3)\) with \(SL_2(\mathbb C)\). He computes reversal as adjugation. His broader mixed-sign carrier permits norm \(\pm 1\), and his displayed axes are not the Pauli axes fixed above.
Here the even center is \(\mathbb R[I]\simeq \mathbb C\), with no nonzero complementary central idempotents. This contrasts with the split compact center \(\mathbb R\oplus \mathbb R\), although both even algebras have real dimension eight. The full algebra \(\mathcal {C}\kern -2pt\ell _{1,3}\simeq M_2(\mathbb H)\) in Example [fcap-0026] is a different algebra. The vector-action image remains the spinor-norm kernel from Theorem [fcap-0023]; no topological component or concrete gamma-map claim is made.