  We describe explicit multiplicative excellent families of rational elliptic
surfaces with Galois group isomorphic to the Weyl group of the root lattices
E_7 or E_8. The Weierstrass coefficients of each family are related by an
invertible polynomial transformation to the generators of the multiplicative
invariant ring of the associated Weyl group, given by the fundamental
characters of the corresponding Lie group. As an application, we give examples
of elliptic surfaces with multiplicative reduction and all sections defined
over Q for most of the entries of fiber configurations and Mordell-Weil
lattices in [Oguiso-Shioda '91], as well as examples of explicit polynomials
with Galois group W(E_7) or W(E_8).
