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A del Pezzo surface of degree one defined over the rationals has 240exceptional curves. These curves are permuted by the action of the absoluteGalois group. We show how a solution to the classical inverse Galois problemfor a subgroup of the Weyl group of type $D_8$ gives rise to a solution of theinverse Galois problem for the action of this subgroup on the 240 exceptionalcurves. A del Pezzo surface of degree one with such a Galois action contains aGalois invariant sublattice of type $D_8$ within its Picard lattice; this canbe characterized in terms of a certain set of sixteen points in $\mathbb{P}^4$.

Sixteen points in $\mathbb{P}^4$ and the inverse Galois problem for del Pezzo surfaces of degree one