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Appendices 1 and 2 to the preprint http://arxiv.org/pdf/alg-geom/9202002v1.pdf by Katz-Morrison, now published in J. Alg. Geom., give nice and explicit generating sets in type $E_6$ and $E_7$. They There is also give a generating set some information on basic invariants in type $E_8$ in Appendix 0, but it is less no explicit formulae (for the obvious reasonreason).
The appendices to the preprint http://arxiv.org/pdf/alg-geom/9202002v1.pdf by Katz-Morrison, now published in J. Alg. Geom., give nice and explicit generating sets in type $E_6$ and $E_7$. They also give a generating set in type $E_8$ but it is less explicit for the obvious reason.