Example. Four real Clifford base entries [lawson2016spin, I.4, Theorem 4.3 and Tables I--II, pp. 27--29] [fcap-0015]
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realCliffordOneZeroEquivProdrealCliffordZeroOneEquivComplexrealCliffordZeroTwoEquivQuaternionrealCliffordOneOneEquivMatrix
L∃∀N

    With \(\mathcal {C}\kern -2pt\ell _{0,0}\simeq \mathbb {R}\) as the scalar anchor, the signature convention of Convention [fcap-0011] gives four nontrivial base entries: \[\begin {aligned} \mathcal {C}\kern -2pt\ell _{0,0}&\simeq \mathbb {R}, & \mathcal {C}\kern -2pt\ell _{1,0}&\simeq \mathbb {R}\times \mathbb {R}, & \mathcal {C}\kern -2pt\ell _{0,1}&\simeq \mathbb {C},\\ \mathcal {C}\kern -2pt\ell _{0,2}&\simeq \mathbb {H}, & \mathcal {C}\kern -2pt\ell _{1,1}&\simeq M_2(\mathbb {R}). \end {aligned}\] For \(\mathcal {C}\kern -2pt\ell _{0,2}\), the two negative generators map to the quaternion units \(i\) and \(j\); their product maps to \(k\). Thus they square to \(-1\) and anticommute, as required by the Clifford relations. The exact generator map is recorded by TauCeti.realCliffordZeroTwoEquivQuaternion_ι.

    For \(\mathcal {C}\kern -2pt\ell _{1,1}\), the positive and negative generators may be represented by \[e_+\longmapsto \begin {pmatrix}1&0\\0&-1\end {pmatrix}, \qquad e_-\longmapsto \begin {pmatrix}0&1\\-1&0\end {pmatrix}.\] Their squares are \(+I\) and \(-I\), and they anticommute. The last equivalence is also the case \(p=q=0\) of Theorem [fcap-0013]. The index swap in Convention [fcap-0011] explains why Lawson's one-generator table lists \(\mathbb {C}\) and \(\mathbb {R}\times \mathbb {R}\) in the opposite order.