Fourth Doctor's High Council

The Fourth Doctor's High Council was the administration formed after the Doctor became the President-elect in unusual circumstances, renaming itself the Supreme Council after realising the one thing it did not possess during the absence of the President was supreme authority. (PROSE: A Brief History of Time Lords)

In the Doctor's absence, Borusa led Gallifrey as Chancellor and Acting President. (TV: The Invasion of Time; PROSE: A Brief History of Time Lords, Doctor Who and the Krikkitmen) The Doctor returned briefly to become officially invested as President and foil the Sontaran invasion of Gallifrey, naming Kelner as Vice President in the process. (TV: The Invasion of Time)

He either resigned after the Sontaran occupation was over which allowed Borusa to become inaugurated as President and form his own High Council (PROSE: A Brief History of Time Lords, The Legacy of Gallifrey) or failed to do so with Borusa spending "aeons" more as Acting President before being awarded the Presidency proper. (PROSE: Doctor Who and the Krikkitmen)

Origins
After the Fourth Doctor returned to Gallifrey on Presidential Resignation Day, Lord President Pandad IV was assassinated. The Doctor was suspected of the crime and invoked Article 17 of the Constitution of the Time Lords as a "guarantee of liberty" to avoid the death sentence. This meant that he was now a candidate in the presidential election to appoint a successor, alongside Chancellor Goth, (TV: The Deadly Assassin) an action which formed an interim High Council. (AUDIO: Mindbomb)

The Doctor eventually uncovered that Goth was the true assassin, acting under the influence of. Goth died shortly afterwards which left the Doctor as the only legible candidate in the election and thus President-elect, (TV: The Deadly Assassin) later becoming President by default. (PROSE: Doctor Who and the Invasion of Time, A Brief History of Time Lords) Despite this, the Doctor continued his travels, leaving Gallifrey without a leader. (TV: The Face of Evil et al.)

Term in office
In the President's absence, and realising that the one thing it didn't have was supreme authority, the High Council renamed itself the Supreme Council. It then "wriggled its way around the technicalities" and nominated Cardinal Borusa as Chancellor and Acting President. (PROSE: A Brief History of Time Lords) The Doctor later called this act "illegal" as his approval was strictly necessary. (TV: The Invasion of Time)

Everyone settled down and looked forward to a few decades of legal wrangling and point-scoring, all certain that the Doctor would not return to assume his position as President, only for him to return to assume his position as President. (PROSE: A Brief History of Time Lords)

As President, the Doctor deactivated the transduction barrier to allow the Vardans to seize control of the Capitol. With the help of Castellan Kelner, whom he named acting Vice President, the Doctor banished those Time Lords dangerous to the new regime to the Drylands. These included political opponents of Kelner including Surgeon General Gomer. (TV: The Invasion of Time)