Base subsets of polar Grassmannians

Let Δ be a thick building of type Xn=Cn,Dn. Let also Gk be the Grassmannian of k-dimensional singular subspaces of the associated polar space Π (of rank n). We write Gk for the corresponding shadow space of type Xn,k. Every bijective transformation of Gk which maps base subsets to base subsets (the shadows of apartments) is a collineation of Gk, and it is induced by a collineation of Π if n≠4 or k≠1.