https://youtu.be/bZPLVDwEr7Y?t=184
He tell a bunch of actors to always 'pick an eye' when in front of a camera. What Michel Cain does not say is that some actors prefer looking out of their 'handed' eye, believing it to have more impact.
As a demo of 'handedness' in eyes, try this experiment:
1. Hold one finger around one foot in front of your face. Hold the other at arms length. Whilst try to look out of both eyes, ensure that the fingers are visually aligned with each other. This will be difficult, as the images from each eye will not correspond.
2. Slowly open and close one eye. Then do the same with the other. The slow blink from one eye (eye A) will not affect the correspondence of the fingers, whilst the slow blink from the other (eye B) will.
The reason is that eye B is adjusting itself to match eye A.
Edit: https://en.wikipedia.org/wiki/Ocular_dominance
Its called ocular dominance apparently.
I thought I understood this at first but with a bit more thought I realised I didn't. You can change the angle between your eyes' directions (i.e. you can cross your eyes rather than them always being totally parallel), so when you focus on single object they are both directly pointing at it simultaneously. Even if your attention is oscillating between them, there's no reason why this would be observable to an outside observer (or camera).
Is it possible you're thinking of looking at someone else's eyes? In that case there are really two objects to look at - the other person's two eyes - so your attention does indeed oscilate in a visible way. But it's not which of your eyes (that you're looking out of) that's varying - it's which of their eyes (that you're looking toward) that's varying.
"The drift of the axes of the eyes was first discovered, and on the whole correctly described, by Dodge (1907). He considered that there is no constant point of fixation and suggested the term 'fixation field." Subsequently, nearly all authors studying eye movements confirmed the presence of drifting movements of the eyes (Glezer and Tsukkerman, 1961)."
I did a bit of work on a project that examined the difference between the way that artists and laymen look at images. For this we used eye trackers. This is where I first saw how restless normal eye movement is. The figures in that chapter and preceding it bear this out.
So... attention drifts from one eye to the other, though one eye is dominant, and defines the 'pan' angle of the other. In addition, generally the eyes scan points within an object rather than look directly at the object as a whole. As someone who teaching drawing (and therefore perhaps analytical looking) this was an 'eye opener'.
Edit: link to referenced book: http://wexler.free.fr/library/files/yarbus%20(1967)%20eye%20...
However, I was reminded of something that I learned when taking drama lessons as a teenager, and which I have used as a teacher. Our attention of an object (such as another human) oscillates from eye to eye. However, it is biased to one eye. Generally this is the one on the opposite site to our writing norm (i.e. left handed = right eyed). If you really want to hold a scene when being filmed, look unwaveringly out of the eye that is nearest the camera. It will feel unnatural, but look dam cool on the screen. Look carefully... you can see Laurence Olivier and others doing precisely this.
As a teacher I use it in class to hold attention. But it’s best effect is on the screen.