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?