They began to show signs of disobedience and left Medina for...
They began to show signs of disobedience and left Medina for Mecca with the alleged aim of making the pilgrimage. They persuaded "the mother of the Faithful" (umm al-mu'minin), A'ishah, who was not friendly with Ali, to join them and in the name of wanting to revenge the death of the third caliph they began the bloody Battle of the Camel. This was done despite the fact that this same Talhah and Zubayr were in Medina when the third caliph was besieged and killed but did nothing to defend him.
Furthermore, after his death they were the first to pay allegiance to Ali on behalf of the immigrants (muhajirun) as well as on their own. Also, the "mother of the Faithful," A'ishah, did not show any opposition to those who had killed the third caliph at the moment when she received the news of his death.
It must be remembered that the main investigators of the disturbances that led to the death of the third caliph were those companions who wrote letters from Medina to people near and far inviting them to rebel against the caliph, a fact which is repeated in many early Muslim histories.
As for the second war, called the Battle of Siffin, which lasted for a year and a half, its cause was the covetousness of Mu'awiyah for the caliphate which for him was a worldly political instrument rather than a religious institution. But as an excuse he made the revenge of the blood of the third caliph the main issue and began a war in which more than a hundred thousand people perished without reason.
Naturally, in these wars Mu'awiyah was the aggressor rather than the defender, for the protest to revenge someone's blood can never occur in the form of defense. The pretext of this war was blood revenge. During the last days of his life, the third caliph, in order to quell the uprising against him, asked Mu'awiyah for help, but the army of Mu'awiyah which set out from Damascus to Medina purposely waited on the road until the caliph was killed.
Then he returned to Damascus to begin an uprising to revenge the caliph's death. After the death of Ali and his gaining the caliphate himself, Mu'awiyah forgot the question of revenging the blood of the third caliph and did not pursue the matter further. After Siffin there occurred the battle of Nahrawan in which a number people, among whom there could be found some of the companions, rebelled against Ali, possibly at the instigation of Mu'awiyah.