A rough classification of symmetric planes (Q2725172)

A stable plane is a pair \((P, {\mathcal L})\), where \(P\) is a locally compact Hausdorff space, whose elements are called points, and \({\mathcal L}\) is system of subsets of \(P\), called lines, such that: \(\bullet\) Every two distinct points \(p, q \in P\) are contained precisely in one line \(p \bigvee q \in {\mathcal L}\) and every line in \({\mathcal L}\) contains at least two points. Besides, there exist three points in \(P\) that don't belong to the same line. \(\bullet\) The covering dimension of \(P\) is positive and finite. \(\bullet\) If two distinct lines \(K\) and \(L\) have a point in common we denote that point by \(K\bigwedge L\). There exists a topology on \({\mathcal L}\) such that the operations \(\bigwedge\) and \(\bigvee \) are continuous, where defined, and the domain of definition of \(\bigwedge \) is an open subset of \({\mathcal L}\times {\mathcal L}\). NEWLINENEWLINENEWLINEA symmetric plane is a stable plane whose point space is a symmetric space such that the symmetries are reflections in the geometric sense. It can have dimension 2, 4, 8 or 16 and the classification problem for dimensions 2 and 4 was solved by \textit{R. Löwen} [Pac. J. Math. 84, 367-390 (1979; Zbl 0426.51010)]. NEWLINENEWLINENEWLINEThe aim of this paper is to give a contribution to the classification of symmetry planes of the remaining dimensions. Every tangent translation plane of a symmetric plane is a Lie triple plane, i.e. a topological affine translation plane whose point space \(M\) is a Lie triple system such that all lines through the origin are subsystems of \(M\) and all inner automorphisms of \(M\) are also automorphisms of the Lie triple plane. In this paper, the author shows that every Lie triple plane is either Abelian or simple or it splits.