A geometric construction of Moufang planes (Q1822760)

Let \(X=(X_+,X_-)\) be a pair of finite-dimensional vector spaces over a field F, let \(N=(N_+,N_-)\) be a pair of cubic forms \((N_+: X_+\to F\), \(N_-: X_-\to F)\) and let T: \(X_+\times X_-\to F\) be a nondegenerated bilinear pairing. The author defines a map {\#}: (X\({}_+,X_-)\to (X_-,X_+)\) and says that (X,N,T) satisfies the adjoint identities if \((x^{\#})^{\#}=N(x)\cdot x\) for \(x\in X_+\) or \(x\in X_-\). For this case some properties are proved and a plane \({\mathcal P}={\mathcal P}(X,N,T)\) is introduced with the point set \(\{x_*=Fx|\) \(x\in X_+\setminus \{0\}:\) \(x^{\#}=0\}\), the line set \(\{y^*=Fy|\) \(y\in X_-\setminus \{0\}:\) \(y^{\#}=0\}\), the incidence \(x_*| y^*:\) \(\Leftrightarrow\) \(x\in \{(y+z)^{\#}- y^{\#}-z^{\#}|\) \(z\in X_-\}\) and the neighbouring \(x_*\approx y^*:\) \(\Leftrightarrow\) \(T(x,y)=0.\) He shows that such examples can be obtained from certain Jordan algebras. In particular he considers Jordan algebras derived from composition algebras A. Then the corresponding plane is a Barbilian plane; and a projective plane if and only if A is a division ring. For \(A={\mathbb{R}}\), \({\mathbb{C}}\) or \({\mathbb{H}}\) the author gives the following geometric construction of these planes: Let \((A^ 4,h)\) with \(h(x,y)=x_ 1\bar y_ 1+x_ 2\bar y_ 2+x_ 3\bar y_ 3+\epsilon x_ 4\bar y_ 4\), \(\epsilon\in \{1,-1\}\) be a unitary space and (\({\mathcal P}_ 3,{\mathcal L})\) the corresponding projective space (\({\mathcal L} = set\) of lines) with the polarity \(\pi\) : \(W\to W^{\perp}\). Let \(Q:=\{x\in P_ 3\), \(x\in \pi (x)\}\) be the corresponding quadratic surface and \({\mathcal S}:=\{\{L,\pi (L)\}|\) \(L\in {\mathcal L}:\) \(L\cap \pi (L)=\emptyset \}\). Then \((P_{\epsilon},{\mathcal S}_{\epsilon})\) with \(P_{\epsilon}:={\mathcal G}_{\epsilon}:=Q\cup {\mathcal G}\) and the incidence \(p| q:\) \(\Leftrightarrow\) \(p=q\) for \(p,q\in Q\); \(p| \{L,\pi (L)\}:\) \(\Leftrightarrow\) \(\{\) L,\(\pi\) (L)\(\}\) \(| p:\) \(\Leftrightarrow\) \(p\in L\cup \pi (L)\); and \(\{\) L,\(\pi\) (L)\(\}\) \(| \{M,\pi (M)\}:\) \(\Leftrightarrow\) \(| M\cap L| =| M\cap \pi (L)| =1\) is the plane looked for. For \(A={\mathbb{H}}\) the plane is a Moufang plane.