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