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

A del Pezzo surface of degree one defined over the rationals has 240 exceptional curves. These curves are permuted by the action of the absolute Galois group. We solve the inverse Galois problem when the del Pezzo surface has a Galois invariant sublattice of type $D_8$ within its Picard lattice. The aforementioned condition can be characterized in terms of a certain set of sixteen points in $\mathbb{P}^4$.