This is the only building you can enter before completing the Mages Guild Recommendations. The door to the plug-in specific Imperial Orrery is across from the entrance. The two doors on either side of it lead to University grounds, which cannot be accessed without the proper authorization. A portal on the east side of the room leads to the Arch-Mage's Tower Council Chambers. The display case beside this door holds three grand soul gems, of which two are filled, and is locked with a very hard lock.
Bothiel and Council member Raminus Polus can be found roaming the lobby during daylight hours. Raminus gives quests and advancements in rank to Mages Guild members, and also sells spells to members and non-members.
The entrance to this room is through a portal in the lobby. A table in the middle of the room holds a scroll of minor heal other and absorb agility can be found on, along with a recipe for a Restore Fatigue potion. Two barrels and a sack, all containing clutter, sit on the east side of the room, with two more clutter sacks in a northern alcove. A display case with a very hard lock beside the barrels holds two replica glass daggers and an empty lesser soul gem. There are two bookshelves in the chamber; a novice alembic sits on top of the southernmost. A portal to the right of the entrance transports members to the Arch-Mage's Quarters. The two barrels respawn, but the three sacks do not.
Council members Irlav Jarol and Caranya can be found in the chambers most of the day.
This is arch-mage Hannibal Traven's sleeping and living quarters. The unique enchanted chest can be found here.
Three bookshelves can be found within the quarters. Three novice retorts and two mortars and pestles can be found on the shelves of one bookshelf and another smaller one holds a few books, including a fragment of the Song of Hrormir, Children of the Sky and The Firmament. Between these two bookshelves stands an enchanting altar. The tallest bookcase contains many books, including Aevar Stone-Singer, Hanging Gardens, More Than Mortal, Provinces of Tamriel, The Seed, Arcana Restored, Spirit of the Daedra, and a novice calcinator. A sample of Nirnroot grows next to this bookcase.
Traven's amenities include two desks, with one holding four books including a copy of Dwemer History and Culture and a copy of The Last King of the Ayleids, as well as a novice alembic and calcinator. A small wine rack lies next to his bed, containing nine bottles of Surilie Brothers Wine and a bottle of Tamika Vintage 399. Between the wine rack and the enchanted chest stands a spellmaking altar. At the foot of his bed a chest of drawers holds clothing, and on the left side of his bed is the Arch-Mage's nightstand, which is the only container in the room safe for long-term storage, and needs a key. A plate with two apples and a pear sits on top. The two baskets at the head of the bed hold a sample of bonemeal and another of imp gall.
Elder Scrolls Wikia entry on this location can be found here.