  We consider the sub-gap physics of a hybrid double-quantum dot Cooper-pair
splitter with large single-level spacings, in the presence of tunnelling
between the dots and finite Coulomb intra- and inter-dot Coulomb repulsion. In
the limit of a large superconducting gap, we treat the coupling of the dots to
the superconductor exactly. We employ a generalized master-equation method
which easily yields currents, noise and cross-correlators. In particular, for
finite inter- and intra-dot Coulomb interaction, we investigate how the
transport properties are determined by the interplay between local and nonlocal
tunneling processes between the superconductor and the dots. We examine the
effect of inter-dot tunneling on the particle-hole symmetry of the currents
with and without spin-orbit interaction. We show that spin-orbit interaction in
combination with finite Coulomb energy opens the possibility to control the
nonlocal entanglement and its symmetry (singlet/triplet). We demonstrate that
the generation of nonlocal entanglement can be achieved even without any direct
nonlocal coupling to the superconducting lead.
