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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.

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