How can I drive this 4 coil motor?
If there's a permanent magnet in the rotor:

Rotate it and record the generated voltage waveforms of the coils. Insert also some load to suppress parasitic oscillations. The you see what's needed for that power which is wasted in the load resistors.

If there's no permanent magnet the case is complex. It can be a reluctance motor or even an induction motor. They are complex. You need an expert to test how they should be run. I do not try to make guesses. Induction motor for ex. needs a proper multiphase AC waveform. The phase shift can be generated by a capacitor but that's not necessary with modern DC to AC conversion electronics. 

But if it has permanent magnet rotor still a basic knowledge of electric machine theory including the math is a must. Without it you cannot decide which simplifications of the waveforms are not disastrous.

You should search for brushless motor drive methods and circuits.