Definition. The hyperbolic plane [lawson2016spin, I.4, Theorem 4.1 and (4.3), pp. 25--26] [fcap-0012]
Definition. The hyperbolic plane [lawson2016spin, I.4, Theorem 4.1 and (4.3), pp. 25--26] [fcap-0012]
The real hyperbolic plane is \[H=(\mathbb {R}^2,Q_{1,1}),\qquad Q_{1,1}(s,t)=s^2-t^2.\] Its coordinate generators \(e_+\) and \(e_-\) satisfy \[e_+^2=1,\qquad e_-^2=-1,\qquad e_+e_-=-e_-e_+.\] For every signature, separating the last positive and negative coordinates gives an isometry \[(\mathbb {R}^{p+q+2},Q_{p+1,q+1}) \simeq (\mathbb {R}^{p+q},Q_{p,q})\perp H.\] The coordinate order is fixed: the retained \(p\) positive and \(q\) negative axes form the first factor, and the last positive and last negative axes form \(H\).