Adjacency preserving mappings of invariant subspaces of a null system

In the space $I_r$ of invariant $r$-dimensional subspaces of a null system in $(2r+1)$-dimensional projective space, W.L. Chow characterized the basic group of transformations as all the bijections $\varphi:I_r\to I_r$, for which both $\varphi$ and $\varphi^{-1}$ preserve adjacency. In the present paper we show that the two conditions $\varphi:I_r\to I_r$ is a surjection and $\varphi$ preserves adjacency are sufficient to characterize the basic group. At the end of this paper we give an application to Lie geometry.