how many injective homomorphism between two lie algebra $sl\_2 $and $sp\_6$ up to conjugate by$Sp\_6$ ?



how many injective homomorphism between two lie algebra sl2 and sp6 up to conjugate by Sp6?
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](http://www.ams.org/mathscinet-getitem?mr=310139)* by Lorente and Gruber. It's of course based on Dynkin, but they work out the details up to rank 6.




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**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 $H$.


