“You can’t tell the players without a program.”
Adonijah was Solomon’s older half-brother. As David’s eldest surviving son, he would normally have been the legitimate heir. But succession in the Bible it always a matter of God’s choice and grace. Adonijah thought he was to be King, but David had other plans, plans he had not exactly shared with everyone. When David’s plan to make Solomon his successor became known, Adonijah seemed to graciously accept the decision, but his request for consolation prize, David’s young nurse (concubine) Abishag, revealed his true heart. To take the wife of a king gave the taker a right to the throne. Bathsheba does not seem to know this. Or, perhaps she did and she knew that by requesting it on behalf of Adonijah she would remove him as a rival to the throne.
Joab of course was David’s cousin and had been David’s most successful general and greatest supporter. But Joab’s ruthlessness turned David’s stomach (nevermind that such ruthlessness probably kept David on the throne). Joab’s support of Adonijah either demonstrates he was not a part of David’s inner-circle, or that he saw himself as a king-maker and determined to subvert David’s wishes. Either way, it cost him his life.
During David’s reign there had been two high priests, both descendants of Eli: Abiathar, and Zadok (though Zadok appears to be older). Abiathar too supported the wrong heir and was removed from the priesthood. From that time on, the family of Zadok determined the High Priesthood.
Frankly, this securing of Solomon’s throne is hardly a tale of righteousness, and I think that’s why the whole story beginning in chapter 2 begins with God’s conditional promise to David: ‘If your heirs take heed to their way, to walk before me in faithfulness with all their heart and with all their soul, there shall not fail you a successor on the throne of Israel.’ The coup described here is pure worldly politics – a foretaste of what is to come.