  Spin-orbit interaction provides a spin filtering effect in carbon nanotube
based Cooper pair splitters that allows us to determine spin correlators
directly from current measurements. The spin filtering axes are tunable by a
global external magnetic field. By a bending of the nanotube the filtering axes
on both sides of the Cooper pair splitter become sufficiently different that a
test of entanglement of the injected Cooper pairs through the Bell inequality
can be implemented. This implementation does not require noise measurements,
supports imperfect splitting efficiency and disorder, and does not demand a
full knowledge of the spin-orbit strength. Using a microscopic calculation we
demonstrate that entanglement detection by violation of the Bell inequality is
within the reach of current experimental setups.
