Do you have sincere feelings for Elizabeth? Does she show interest in your friend? If you answered yes to the first and no to the second, I would talk to your friend to give him a heads-up and then ask Elizabeth out. It sounds like she is the person you want to date and the only thing in the way is your friend's unrequited feelings for her. As sad as it is for him that he has feelings for a woman who isn't interested in him, that doesn't give him any rights and doesn't make you and her any less free of pursuing your own mutual feelings for one another. If you don't pursue Elizabeth, she will eventually move on and pursue someone else anyway.

When I first started dating my husband, we had a common friend who was head over heels for me. I'm really glad that my husband didn't let this stop him. And for what it's worth, it didn't destroy the his or my friendship with the other dude. My husband was upfront with him and gave him a heads-up and the other dude got over it.

This is assuming she'd be the one you'd pick if it weren't for your friend and that you do have some feelings for her. If it's not the case, then Alicia.