The author continues his work on a classification program for stable planes \(\mathbf E\) with automorphism groups of large dimension. A stable plane is a topological linear space such that the set of pairs of intersecting lines is open. The point set \(M\) is assumed to be locally compact and of finite positive covering dimension. A deep result of R. Löwen [J. Reine Angew. Math. 343, 108-122 (1983; Zbl 0524.57011)] shows \(\dim M \in \{ 2, 4, 8, 16\}\). Prominent examples are the open subgeometries of the projective planes over the reals, the complex numbers, the quaternions and the octonions.
The group \(\Sigma\) of automorphisms of a stable plane becomes a locally compact group of finite dimension when endowed with the compact-open topology. The paper under review studies almost simple subgroups \(\Delta \leq \Sigma\) in the case \(\dim M = 8\). The main result is that if \(\dim \Delta\) strictly exceeds \(16\), then \(\Delta\) is isomorphic to PSL\(_3 ({\mathbf H})\) or PU\(_3 ({\mathbf H}, 0)\) or PU\(_3 ({\mathbf H}, 1)\). The possible actions of these so-called “classical motion groups” on \(8\)-dimensional stable planes are known [cf. R. Löwen, Result. Math. 9, 119-130 (1986; Zbl 0588.51018)]: In any case, \(\mathbf E\) is isomorphic to an open subplane of the projective quaternion plane P\(_2 \mathbf H\) and the action of \(\Delta\) on \(\mathbf E\) is induced by the natural action of \(\Delta\) on P\(_2 \mathbf H\). By contrast, there is an action of the \(16\)-dimensional group SL\(_3 {\mathbf C}\) on the \(8\)-dimensional Hughes planes [see M. Stroppel, Can. Math. Bull. 37, No. 1, 112-123 (1994; Zbl 0801.51013)].