Is there a classification of embeddings of SL\_2 into SP\_6 as algebraic groups over Q and R respectively? 


see also the link:mathoverflow.net/questions/36762,



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](http://books.google.com/books?isbn=9780486638324).


