Example. The real Clifford algebra of signature (1,3) [fcap-0026]

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realClifford_classification_of_residue_eq_two

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Use the convention of Convention [fcap-0011]: \(p\) counts positive squares, \(q\) counts negative squares, and a Clifford generator obeys \(v^2=Q(v)1\). Then the real algebra for one positive and three negative axes is \[\mathcal {C}\kern -2pt\ell _{1,3}\simeq M_2(\mathbb H).\] The Tau Ceti classification theorem states existence of an abstract real algebra equivalence. Here \(q-p=2\) is its residue modulo \(8\), and its matrix size is \(2^{(p+q-2)/2}=2\).

[chevalley1954algebraic, II.2.9, pp. 65--66] uses the same positive/negative index convention. Split off one hyperbolic plane of signature \((1,1)\); the orthogonal remainder is a negative plane of signature \((0,2)\). Chevalley's dimension-two definite calculation on p. 66 identifies its Clifford algebra with \(\mathbb H\). The hyperbolic-plane theorem [chevalley1954algebraic, II.2.1, pp. 42--43] gives \(\mathcal {C}\kern -2pt\ell _{1,1}\simeq M_2(\mathbb R)\).

[chevalley1954algebraic, II.2.5, pp. 45--46] gives an orthogonal-sum tensor-product formula for an even-dimensional nondegenerate first summand; the second quadratic form is twisted. Here normalized hyperbolic generators \(e^2=1\) and \(f^2=-1\) have volume \(J=ef\) with \(J^2=1\). For a vector \(v\) in the orthogonal remainder, \(Jv\) commutes with \(e,f\) and still squares to \(Q(v)\). Since \(v=J(Jv)\), these commuting factors generate the full algebra. Thus the special hyperbolic split leaves that remainder unchanged: \[\begin {aligned} \mathcal {C}\kern -2pt\ell _{1,3}&\simeq \mathcal {C}\kern -2pt\ell _{1,1}\otimes _{\mathbb R}\mathcal {C}\kern -2pt\ell _{0,2}\\ &\simeq M_2(\mathbb R)\otimes _{\mathbb R}\mathbb H \simeq M_2(\mathbb H). \end {aligned}\] Both sides have real dimension \(16=2^4=4\dim _{\mathbb R}\mathbb H\). By comparison, \(M_4(\mathbb C)\) has real dimension \(32\). This abstract classification alone does not identify any particular gamma-matrix map, its kernel, or its image; the concrete Dirac map is treated in Proposition [fcap-002U].