What is the generator of translation in the BeltramiKlein model of the hyperbolic plane?

closed as too localized by Ryan Budney, José FigueroaO'Farrill, Will Jagy, Richard Kent, Andy Putman Jun 3 '11 at 4:45
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Hyperbolic plane is a homogeneous space $G/H$ where $G$ acts by isometries and so any reasonable defined `translation' is given by the natural left action of $G$. Decide what elements of $G$ will you call translations and then for any abstract generator $X\in\mathfrak{g}$ such that $\exp{tX}$ is translation, compose the natural left action with diffeomorphism of $G/H$ to your favorite model. Differentiate the resulting map at zero and you are done. If I am not mistaken, this question is more suitable for math.SE. 

