A characterization of the Grassmannian of points and lines for \(C_{3,2}\)-buildings (Q1346378)

Let \(\Delta\) be a building of certain type \(X_ n\) and let \(i\) be a certain node of the diagram. Then the point-line space of type \(X_{n,i}\) corresponding to \(\Delta\) is the point-line geometry whose points are the varieties of type \(i\) of \(\Delta\) and whose lines are the sets of varieties of type \(i\) lying in a flag of cotype \(\{i\}\) of \(\Delta\). A point-line characterization of the class of buildings of type \(X_ n\) relative to a certain node \(i\) of the diagram is giving necessary and sufficient conditions for a point-line geometry to be a point-line space of type \(X_{n,i}\) corresponding to a certain building of type \(X_ n\). For spherical buildings all but a finite number of types \(X_{n,i}\) have been characterized, except for the infinite series \(C_{n,n - 1}\). The present paper covers the first element in that series, namely, the author characterizes spaces of type \(C_{3,2}\) and calls them cuboctahedral spaces. The main axioms describe the intersection possibilities for subspaces isomorphic to a projective plane or a generalized quadrangle. In the meantime, the author has done the job for all \(C_{n,n - 1}\) spaces.