Alex is a nickname for Alexander. Some other nicknames for it are Xander and Lex- so I guess Lex would work as a nickname for Alex. I've never heard Al as a short version of either name, but I guess toi could use it. But people will think it's short for Albert.
I lovingly call my Alex, short for Alessandro .. I call him Aleoooop ou if we are being silly call him Aleien (alien). :) its easy to give someone a nickname after toi have rely known the person. Many nicknames come from secret inside jokes ;)