Can i put retinol on eyelids and region under eye for dark circles?

The dark circles are often a reflection of slow blood flow in that area due to allergies or other nasal issues. Nothing you apply to that skin surface would alter that deep blood flow. I would check with your doc about the origin of this issue before trying this.

It probably depends on the formulation. I suggest checking the package instructions and warnings about use near the eye. In any event, you should avoid getting any non-ophthalmic medication in your eyes.