Robin is correct, the splitting comes from the two 3p levels, which would be degenerate (i.e. no doublet) if not for the spin-orbit coupling – a sort of internal Zeeman effect from the magnetic field induced by the orbiting electron. This splits the levels and creates the doublet. If we apply a weak magnetic field (similar in size to the spin-orbit bit) we’ll get Zeeman splitting of these levels and end up in turn with 10 lines (5 from each line of the doublet)
The Na triplet is more complicated, but I think this comes from a similar 4s -> 3d transition, and selection rules exclude some of the 5 sublevels of the 3d orbital, giving a triplet – I’m slightly less clear on this one.