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A Wiki of Ice and Fire: Robert's Rebellion
The conflict was instigated after Rhaegar Targaryen had kidnapped Lyanna Stark, who was betrothed to Robert Baratheon. Despite the pleading of Hoster Tully for patience, her brother Brandon Stark rode to the city of King's Landing. There, he went into the Red Keep shouting for Prince Rhaegar to come out and die. King Aerys II arrested him and his companions (his squire Ethan Glover, Jeffory Mallister, Kyle Royce, and the nephew and heir of the Lord of the Eyrie Elbert Arryn) for threatening and conspiring to murder the crown prince.
King Aerys II, who had become increasingly paranoid, since his several months of confined imprisonment during the Defiance of Duskendale, had ordered the fathers of the prisoners to come south to answer the charges against their sons. When they did, they and their sons were all murdered without trial. Ethan Glover was the only survivor. Lord Rickard Stark who demanded a trial by combat was roasted in his armor, as his son Brandon Stark was put in a strangulation device and was forced to watch, strangling himself in the process of trying to save his father. Soon after, the Mad King demanded the heads of Robert Baratheon and Eddard Stark from their guardian, Jon Arryn, Lord Arryn of the Eyrie. Rather than comply, Lord Arryn raised his banners in revolt against these unjust acts.
When Lord Arryn raised his banners in revolt, many of his bannermen sided with the King, chief among them Lord Grafton, who called the other loyalists to aid him in barring the rebels from entering the port of Gulltown. Eddard Stark decided to make his own way back to the North via the Fingers. Gulltown's defenses eventually fell and Robert was able to return to Storm's End. Both men then called on their own bannermen.
Next House Tully joined the war on the rebels' side, following the marrige of Eddard and Jon to Catelyn and Lysa respectively, binding House Stark, Tully and Arryn togther. However, like in the Vale, not all the Tully bannermen joined the rebel cause. Houses Ryger, Darry, and Mooton took the side of the Targaryens in the war and Frey remained neutral until the situation became clearer.
After all the rebels were united, it was decided they would stand behind Robert, whose grandmother had been the daughter of King Aegon V Targaryen, giving him the best claim to the throne outside of Aerys, his children and grandchildren.
In turn, King Aerys summoned his own banners. All Targaryen bannermen responded as did House Tyrell and Martell who stayed loyal to the throne as did all of their bannermen, including Redwyne and Hightower. However, House Lannister, led by Tywin Lannister, who had recently resigned his post as Hand of the King ignored the summons, while House Greyjoy decided to remain neutral throughout the conflict.