Why fast eye movement splits a white LED into the RGB components? I have a "white" LED as the notification light on my phone. When I look at it straight it looks whitish, but when I move my eyes around (or shake my head) I suddenly see glimpses of red and green (and probably blue) lights coming out of there. Why?
 A: There are most likely three LEDs with different colors in that device which are driven by current pulses of different length and timing to achieve an arbitrary RGB color with the least amount of hardware. The result is that you are seeing a stroboscopic effect when you move your eyes fast enough to resolve the timing of the three colors. This is also known as persistence of vision, and the educational channel Veritasium covered it in this video, showing it in action.
If it's just a white LED with phosphors illuminated by a blue LED as in the answer by count_to_10, it could be the different phosphorescence times of the phosphors after a short pulse excitation by the one blue LED chip. 
In either case the eye movement turns the temporal variation of the color into a spatial variation. While your eye is insensitive to the temporal variation (the designer of the device has chosen the "chopping" frequency just high enough to be invisible), it is quite sensitive to spatial variation. So what you are basically doing is to move different "frames" of the light to different locations on your retina.
A: If you coat a blue LED   with a phosphor material, some of  the blue light will be changed to red, green and yellow light, (which is intended to be seen as white light) and maybe, and I don't know how this could happen, I admit,  shaking your head might allow you to resolve these components. 

I originally had a biology element to this answer, but CuriousOne's comments above  have brought me back to a physics based explanation, and this edit is to reflect that. 

