Ruah Brackett was a Starfleet officer.
Brackett attended Starfleet Academy at the same time in the 2320s as Jean-Luc Picard, who she had a friendly rivalry with during debates held by the debating team. Brackett also knew Miles O'Brien. (TNG novelization: Unification)
Brackett, then a commander, was a professor at Starfleet Academy in the 2350s and taught mathematics. Cadet Kathryn Janeway, who wanted to take Brackett's differential geometry course during her Academy senior year, once came upon a dog owned by Brackett called Chomel. Brackett had adopted the dog after finding him in Golden Gate Park. (VOY novel: Mosaic)
After Ambassador Spock was discovered to be on Romulus in 2368, Admiral Brackett urgently ordered Jean-Luc Picard and the USS Enterprise to Starbase 234 two days later to give Picard the information with the intention of having him investigate. (TNG episode: "Unification I")
After Picard visited Sarek and was told that Spock was acquainted with the Romulan Senator Pardek, Admiral Brackett gave Picard permission to cross the Romulan Neutral Zone and go to Romulus, although she wanted to know as little about his plan as possible. Picard knew Brackett would deny involvement if anything went wrong on the mission. (TNG novelization: Unification)
Several days later, Admiral Brackett assigned the crew of the USS Hood to represent Starfleet at a memorial service to honor Sarek on Vulcan and also allowed Picard and the Enterprise crew to attend as well. Admiral Brackett questioned that Spock's "cowboy diplomacy" would work to bring about reunification. (TNG short story: "Last Words")