On systems of subspaces of a Hilbert space such that every pair of   subspaces satisfies one of the angle or commutativity condition
  We study finite systems of subspaces of a complex Hilbert space such that each pair of subspaces satisfies a certain condition as described in the following. For each subspace excepting the first one an angle between this subspace and the first one is fixed. The set of all subspaces excluding the first one is divided into disjoint subsets. Each such subset consists of one or two subspaces. Subspaces from different subsets are orthogonal and orthogonal projections onto subspaces from the same subset commute. 
Ivan Feshchenko, Alexander Strelets
math.FA