Example. The split three-dimensional Spin group is \(SL_2(\mathbb R)\) [fcap-002I]

✍️sourceAGENTDRAFTED

For \(Q_{2,1}(x_0,x_1,x_2)=x_0^2+x_1^2-x_2^2\), set \(A=e_0e_1\), \(B=e_0e_2\), and \(C=e_1e_2\). Then \[A^2=-1,\qquad B^2=C^2=1,\qquad AB=-C.\] The assignments \[A\longmapsto \begin {pmatrix}0&1\\-1&0\end {pmatrix},\qquad B\longmapsto \begin {pmatrix}0&1\\1&0\end {pmatrix},\qquad C\longmapsto \begin {pmatrix}-1&0\\0&1\end {pmatrix}\] satisfy these relations, and the images of \(1,A,B,C\) form a basis of \(M_2(\mathbb R)\). They therefore define an algebra equivalence \(\Phi \) with \[\Phi (s+aA+bB+cC)= \begin {pmatrix}s-c&a+b\\-a+b&s+c\end {pmatrix}.\] Reversal negates \(A,B,C\), so it becomes matrix adjugation, and \[N_{\mathrm {rev}}(s+aA+bB+cC) =s^2+a^2-b^2-c^2=\det \Phi (s+aA+bB+cC)\] CliffordAlgebra.map_reverseEven_eq_adjugate_of_finrank_eq_three CliffordAlgebra.reverseEven_mul_eq_one_iff_det_eq_one_of_finrank_eq_three.

Restriction gives \(\operatorname {Spin}(Q_{2,1})\to SL_2(\mathbb R)\). Conversely, if \(M=(m_{ij})\in SL_2(\mathbb R)\), put \[\begin {aligned} s&=\frac {m_{00}+m_{11}}2,& a&=\frac {m_{01}-m_{10}}2,\\ b&=\frac {m_{01}+m_{10}}2,& c&=\frac {m_{11}-m_{00}}2. \end {aligned}\] Then \(\Phi (s+aA+bB+cC)=M\), and its reverse norm is \(\det M=1\). The carrier lemma puts this inverse value in the actual Spin group; the algebra maps give two-sided multiplicative inverses. Thus \[\operatorname {Spin}(Q_{2,1})\simeq SL_2(\mathbb R).\]

The source translation matters. Lawson--Michelsohn's convention turns this even algebra into \(\operatorname {Cl}_{0,2}\simeq M_2(\mathbb R)\) via [lawson2016spin, I.3, Theorem 3.7, pp. 23--24; I.4, (4.0), p. 25]. With the reversed index convention, [gallier2014clifford, Example 1.3, p. 33] independently records \(\operatorname {Cl}_{2,0}\simeq M_2(\mathbb R)\), the same real matrix base algebra. Their indefinite Pin/Spin carrier conventions allow factors of quadratic value \(\pm 1\) [lawson2016spin, I.2, (2.24)--(2.25), p. 18] [gallier2014clifford, Section 1.5, Definition 1.6, p. 31]. We use their algebra identifications but impose the displayed reverse-norm-one equation. For example, \(e_0e_2\) has reverse norm \(-1\) and maps to a determinant \(-1\) matrix, so it is excluded here.

This split form has polarization \(W=\mathbb R(e_0+e_2)\), \(W'=\mathbb R((e_0-e_2)/4)\), and line \(\mathbb R e_1\): the unhalved polar pairing of the displayed generators is \(1\), and the line coordinate squares to \(Q_{2,1}\). TauCeti's polarized dimension-three theorem therefore gives an independent abstract group equivalence with \(SL_2(\mathbb R)\) TauCeti.exists_spinGroup_mulEquiv_specialLinearGroup_and_spinRep_equiv_stdSLRep_of_finrank_eq_three. The concrete map \(\Phi \) above is not asserted to be definitionally that map, and the theorem's representation refinement belongs to the later module packet. Likewise, the vector-action image is the spinor-norm kernel from Theorem [fcap-0023], not all of \(SO(2,1)\). No component or covering-space claim is made.