For a small burn scar under the eye, silicone-based scar gels are generally the most effective option, as they help to soften and gradually reduce its appearance over time. Apply a very thin layer once or twice daily, taking care to avoid direct contact with the eye area, as the skin there is delicate. Using sunscreen every day is also important, as it helps prevent the scar from becoming darker. If there is no improvement after a few months, or if the scar is very close to the eye, please consult a dermatologist for further treatment options.