On the Grassmann space representing the lines of an affine space

In 1982, Bichara and Mazzocca characterized the Grassmann space G r ( 1 , A ) of the lines of an affine space A of dimension at least 3 over a skew-field K by means of the intersection properties of the three disjoint families Σ 1 , Σ 2 and T of maximal singular subspaces of G r ( 1 , A ) . In this paper, we deal with the characterization of G r ( 1 , A ) using only the family Σ = Σ 1 ∪ Σ 2 of maximal singular subspaces.