  We give a proof for the Kato square root problem on suitable open sets and
subject to pure Dirichlet boundary conditions. In contrast to earlier work, our
proof uses only second-order techniques and is in the spirit of the proof given
by Egert--Haller--Monniaux--Tolksdorf in the scope of the 27th international
internet seminar for $\Omega = \mathbb{R}^n$. In contrast to the usual proofs
based on the first-order framework by Axelsson--Keith--McIntosh, the proof
becomes substantially shorter and technically less demanding.
