What is the ruling on looking at the hand of an elderly female teacher without lust, and is that considered a necessity?

Islamic law has prohibited the mixing of men and women (ikhtilat) and has blocked all means leading to it. This includes mixed private lessons, unless they adhere to the legitimate Islamic guidelines. An exception is made if there is a dire need and no alternative is available, provided the legitimate Islamic guidelines are still observed. As for looking at a woman's face and hands, it is forbidden according to the relied-upon opinion, unless she is an old woman for whom there is no desire.