Well in his/her/fox's case, the name Lily belongs to another person.
I'd suggest taking a peek at this
naming rules on the wiki.
You might ask @main if there's currently a famous Lily around.
(I can't remember one, but heh~)
If he/she isn't active anymore, using LiIy seems okay. So long as you don't go out of your way pretending to be Lily the 1st (But of course you won't do such a thing, right, right?)
[Edit: holy mackerel, two ninjas!]