That quest just sits there but there's no way to ask Gaelius about it or reach any library. Anyone found a way to it?

You need to be an Aurelian Praetor or a thief in TG6.

How one can become an Aurelian Praetor? Does it mean it can be achieved by starting a game as a Daratan Praetor and then betraying them in favor of Aurelian? Or a simple pledge of loyalty to Aurelian will be enough at the end?

What is "TG6"??? Thieves Guild, mission 6 (last mission in Maadoran)?

I must admit that your response is a bit vague

I don't know for sure whether or not you have to start as a Daratan Praetor. It's possible you can get promoted if you do enough to help the Aurelians.

Only Aurelian Praetors and thieves can access the library to complete the map quest.

Yes TG6 is thieves guild mission 6. I think it is the last one in Maadoran.

As for how to become an Aurelian Praetor: You don't need to start as a Daratan Praetor. It's possible to join the Aurelians as any starting character, although how you go about this will be different depending on faction membership (or lack thereof). In some cases (such as if you're a Daratan Praetor) you will become an Aurelian Praetor immediately, in other cases you have to "earn his trust" as a Liegeman first.

In most cases, you will have to betray your current faction in order to join the Aurelians. This is usually presented as an obvious choice, but if you're already in Maadoran, and you have some potentially juicy information for Gaelius, just visit him and spill the beans.

It's also possible to join Gaelius from Teron without betraying your faction (if you have one - although you'll still lose your faction membership). Since this leads to a mission (and location) that you won't even hear about otherwise, I won't spoil this here (unless you specifically want me to). It's definitely worth doing this at least once - you'll miss out on most of your "standard" faction missions (and it will close off most of Teron's missions, since you'll move to Maadoran), but instead you'll gain a couple of extra Aurelian missions to make up for it.

If you're faction-less or part of an Aurelian-friendly faction (and haven't done any missions in Maadoran for the faction yet) you can also get the above mission/location by just asking Gaelius nicely.

Yes you can - it's exactly as you described for the IG. Which is what I meant when I wrote: "In some cases (such as if you're a Daratan Praetor) you will become an Aurelian Praetor immediately, in other cases you have to "earn his trust" as a Liegeman first."