arithmetic geometry - Is there a classification of embeddings of SL_2 into SP_6 as algebraic groups over Q and R respectively?

You probably want the Jacobson–Morozov theorem, which says that homomorphisms of the Lie algebra sl₂ over a field of characteristic 0 to a semisimple Lie algebra g can be classified in terms of the nilpotent elements of g. More precisely, if e, f, h, is the usual basis of sl₂ then you can choose the image of e to be any nilpotent element of g, and the images of f and h are then determined up to conjugation by the centralizer of e.

For details see Jacobson's book on Lie algebras.