  We study the self-adjointness of the two-dimensional Dirac operator coupled
with electrostatic and Lorentz scalar shell interactions of constant strength
$\varepsilon$ and $\mu$ supported on a closed Lipschitz curve. Namely, we
present several new explicit ranges of $\varepsilon$ and $\mu$ for which there
is a unique self-adjoint realization with domain included into
$H^{\frac{1}{2}}$. A more precise analysis is carried out for curvilinear
polygons, which allows one to take the corner openings into account. Compared
to the preceding works on this topic, two new technical ingredients are
employed: the explicit use of the Cauchy transform on non-smooth curves and the
explicit characterization of the Fredholmness for singular integral operators.
