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Question: Given a twist of the projective space, how do I find unit quaternions that represent it?

Backgroud and what do I mean: Following Conway & Smith's On Quaternions and Octonions, every pair of unitary quaternions $l,r$ gives a map $[l,r]:x \mapsto \bar{l}xr$, and a map $*[l,r]:x \mapsto \bar{l}\bar{x}r$, and every element of $GO(4)$ can be seen as one of these two mappings (and $SO(4)$ consists of those of the first type). When working projectively, the four expressions $[\pm l, \pm r]$ are all the same projective map.

On the other hand, the elements of $PSO(4)$ can be thought, in a geometric way, as twists, that is, the product of two rotations on polar lines with different angles (in particular a rotation happens when one of the two angles is zero).

I would like to see, geometrically, what is the twist obtained by the product of two twists that I know (in terms of their lines). Using quaternions that would be straightforward, if I knew, given a (geometric) twist (ie. its two polar lines and the angles) how can we find the unit quaternions $l,r$ that represent it?

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closed as off topic by Qiaochu Yuan, Ryan Budney, Mariano Suárez-Alvarez, Willie Wong, S. Carnahan Feb 20 '11 at 15:03

Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here.If this question can be reworded to fit the rules in the help center, please edit the question.

This is not really an appropriate question for MO; perhaps you should ask it on instead. – Qiaochu Yuan Feb 20 '11 at 0:53
I second Qiaochu's suggestion. It is an interesting question, but this is not the proper home. – S. Carnahan Feb 20 '11 at 15:04