Vanishing of the Brauer group of a del Pezzo surface of degree 4

We explicitly construct a del Pezzo surface $X$ of degree 4 over a field $k$ such that ${\mathrm{H}}^1(k,\mathrm{Pic}\bar{X})$ is isomorphic to $\mathbb{Z}\mathbb{Z}\mathrm{/}2\mathbb{Z}$ while $\mathrm{Br}X\mathrm{/}\mathrm{Br}k$ is trivial. This proves that the algorithm to compute the Brauer group in [VAV] cannot be generalized in some cases.