Geometry of skew-Hermitian matrices

Let D be a division ring with an involution ¯. Assume that F={a∈D:a=a¯} is a proper subfield of D and is contained in the center of D. Let SHn(D) be the set of n × n skew-Hermitian matrices over D. If H1,H2∈SHn(D) and rank(H1 − H2) = 1, H1 and H2 are said to be adjacent. The fundamental theorem of the geometry of skew-Hermitian matrices over D is proved: Let n ⩾ 2 and A be a bijective map of SHn(D) to itself, which preserves the adjacency. Then A is of the form A(X)=αtP¯XσP+H0∀X∈SHn(D), where α ∈ F*, P ∈ GLn(D), H0∈SHn(D), and σ is an automorphism of D.