What is the ruling on touching the eye's water with the hand, and taking some water with the hand, without washing the hand from the used water that has mixed with the eye's water and the sweat of another limb?

Used water is not sufficient for purification according to most scholars, but it is pure. So, if your hand touches water used on the face, eye, or any other limb, you are not required to wash your hand, and it is permissible for you to take water with it to wash any limb of the ablution.