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In this article Dirac operators $A_{\eta, \tau}$ coupled with combinations ofelectrostatic and Lorentz scalar $\delta$-shell interactions of constantstrength $\eta$ and $\tau$, respectively, supported on compact surfaces $\Sigma\subset \mathbb{R}^3$ are studied. In the rigorous definition of theseoperators the $\delta$-potentials are modelled by coupling conditions at$\Sigma$. In the proof of the self-adjointness of $A_{\eta, \tau}$ a Krein-typeresolvent formula and a Birman-Schwinger principle are obtained. With theirhelp a detailed study of the qualitative spectral properties of $A_{\eta,\tau}$ is possible. In particular, the essential spectrum of $A_{\eta, \tau}$is determined, it is shown that at most finitely many discrete eigenvalues canappear, and several symmetry relations in the point spectrum are obtained.Moreover, the nonrelativistic limit of $A_{\eta, \tau}$ is computed and it isdiscussed that for some special interaction strengths $A_{\eta, \tau}$ isdecoupled to two operators acting in the domains with the common boundary$\Sigma$.

On Dirac operators in $\mathbb{R}^3$ with electrostatic and Lorentz scalar $\delta$-shell interactions