# An attempt to improve Plücker coordinates

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)

$$u=d+m$$

$$v=d−m$$

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

$$uq=qv$$

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

$$u⋅u′=v⋅v′$$

and their common point is

$$u(u′−v′)+(u′−v′)v$$

if the moments are not collinear.