I think "Fraternally" would be appropriate, as would "Best Wishes," "Sincerely," "Respectfully," etc., for formal correspondence related to Greek Life with those outside your GLO.
If you're writing to a fellow Greek and it doesn't have to do with a Greek issue, don't sign "Fraternally." Use an appropriate formal closing, like the others I mentioned above.
Writing an informal note to another Greek, I might sign it "PanLove," "Greek Love" or use "Pi Love," "Pi Love and Mine," etc., depending on who I was talking to!
In ADPi Speak, I would close a formal letter to a sister with "Loyally." If it was to a sister I am close to I would close it "Love & Loyally" or "L&L." A informal letter to an ADPi would be closed with "Pi Love" or "Pi Love & Mine" (aka PLAM).
Good topic! No one seems to write letters anymore and it's good to think about this stuff! Makes me want to write to people!