Orthogonality preserving bijective maps on finite dimensional projective spaces over division rings

Let nu be a finite dimensional right vector space over a division ring , equipped with a nondegenerate semi-sesquilinear inner product <.,.>. Let D and E be bijective linear operators on nu. The operators D and E induce indefinite inner products on nu, by the formulas < Dx,y > and < Ex,y >, where x,y is an element of nu. We study maps on the projective space over nu that send D-orthogonal one-dimensional subspaces (elements of the projective space) to E-orthogonal one-dimensional subspaces. We prove that under the assumption of bijectivity of the map, and assuming n 3, such a map T preserves (D, E )-orthogonality if and only if it preserves (D, E )-orthogonality in both directions. In this case, it is induced by a semilinear transformation on nu that is (D, E )-unitary up to a multiplicative constant. For n = 3, it turns out the injectivity assumption alone suffices to reach the same conclusion. The main results are specialized to the particular cases when is the (commutative) complex field and the (noncommutative) skew field of real quaternions. In particular, the existence of (D, E )-unitary semilinear maps is discussed in these cases.