The A ring is composed of several rooms. It has several storage rooms, the Brotherhood Library and Archives, The Den and the Council Room.
The Library and Archives store knowledge the Brotherhood collected. Also, Scribe Jameson collects the holo-tags from fallen Brotherhood brothers and sisters.
The Lyons' Pride mostly spends time in the room named as "The Den". The place is certainly distinctive from the others, as it has more food than the other areas in A Ring. Here Lyons' Pride members can relax and discuss tactics about defeating the super mutants and the Enclave.
The Great Hall is actually one of the most important buildings in the Citadel. Here Elder Owyn Lyons discusses the situation with other Paladins and Knights.