how many injective homomorphism between two lie algebra sl2 and sp6 up to conjugate by Sp6? how many injective homomorphism between two lie algebra $sl_2 $and $sp_6$ up to conjugate by$Sp_6$ ?
 A: To follow up Skip's comment, assuming the underlying field is algebraically closed of characteristic 0, the number of possible embeddings (= injective homomorphisms) of $\mathfrak{s}\mathfrak{l}_2$ into $\mathfrak{s}\mathfrak{p}_6$ up to conjugacy by the adjoint group should be 7.   This is the number of nonzero nilpotent conjugacy classes in the simple Lie algebra $C_3$, which in turn are in natural bijection (using Jacobson-Morozov theory) with the $\mathfrak{s}\mathfrak{l}_2$-triples.   A good source for the basic theory is Section 11 of Chapter 8 in the Bourbaki treatise Groupes et algebres de Lie
(published in English by Springer), supplemented by data on hilpotent orbits in books like those by Collingwood-McGovern and Carter.    As Skip points out, general questions of this sort were studied systematically by Dynkin and later refined or generalized by others.   
A: As a follow-up to Jim's answer (which came in as I was typing an inferior answer), let me add that the 7 possible embeddings are given in the $C_3$ entry of Table VI in the paper: Classification of semisimple subalgebras of simple Lie algebras by Lorente and Gruber.  It's of course based on Dynkin, but they work out the details up to rank 6.

Added
The defining vectors for the 7 embeddings are given by: (1,0,0), (1,1,0), (1,1,1), (2,2,0), (3,1,0), (3,1,1) and (5,3,1).  Recall that the embedding with defining vector $(a,b,c)$ is one for which the Cartan generator $H$ of the $\mathfrak{sl}(2)$ subalgebra is given by $H = a H_1 + b H_2 + c H_3$, where $(H_i)$ is an orthonormal basis of a Cartan subalgebra of $C_3$ containing $H$.
