Consider a 2-dimensional subspace of 4-dimensional vector space (of quaternions). It is a line in projective 3-space $P^3$. Let $u,v$ be the following 3-vectors (or vector quaternions)
where $d,m$ are Plücker coordinates of this subspace. They have the same length (because $d$ and $m$ are orthogonal). I can prove that a quaternion $q$ belongs to the subspace iff
Is there a simple (or not very complicated) formula for line-line meet in such coordinates? I have a formula for the case of not collinear moments, but not in general case. Two lines $u,v$ and $u′,v′$
are coplanar iff
and their common point is
if the moments are not collinear.