To any two-dimensional rational plane in four-dimensional space one can
naturally attach a point in the Grassmannian Gr(2,4) and four lattices of rank
two. Here, the first two lattices originate from the plane and its orthogonal
complement and the second two essentially arise from the accidental local
isomorphism between SO(4) and SO(3)xSO(3). As an application of a recent result
of Einsiedler and Lindenstrauss on algebraicity of joinings we prove
simultaneous equidistribution of all of these objects under two splitting
conditions
