My question is this: Is the projective plane $P(K^3)$ (the points are the one-dimensional subspaces and the lines are the two-dimensional subspaces) for a division ring $K$ isomorphic to its dual?
I know that this is true if $K$ is a field, and I also see several ways to prove it (by using the dual vector space or a bilinear form). However, none of these proofs are in an obvious way still correct if $K$ is only a division ring.
Thus, I am not at all sure how to prove the above statement (if it is correct at all) without running into some problem with non-commutativity. Can anybody help?