I am interested in an explicit description of the principal homomorphism from $SL(2,\mathbb{C})$ to $G$, for each complex semisimple Lie group $G$. Does any one have specific references please? Kostant's original paper is of course great, and contains a lot, but I do not think it contains explicit descriptions for each specific example (which is what I really want at the moment).



Where can I find explicit descriptions of principal $SL(2,\mathbb{C})$s?
You may try exploring the reference 


"Lie Algebras, Geometry, and Toda-Type Systems" 


by Alexander V. Razumov, Mikhail V. Saveliev, Cambridge University Press.


