In 2:12-46 we see that Solomon ended the factors of rebellion for the establishment of his kingdom.
Adonijah attempted to usurp the throne of David and was subdued and pardoned by Solomon (1:5-10, 41-53). After some time he asked Solomon to give him a certain woman as his wife, and Solomon rejected him and sentenced him to death (2:13-24). Solomon used that occasion as a chance to end Adonijah's life.
Because Abiathar had joined the rebellion of Adonijah, Solomon discharged him from the post of high priest (vv. 26-27).
In verses 28 through 34 Solomon sentenced Joab to death. Joab was the captain of the army who helped Adonijah's usurping of the throne of David and who killed the innocent captains of the armies, Abner and Amasa (1:7; 2:32). Solomon executed Joab according to his father David's charge (vv. 5-6, 28-34).
Next, Solomon changed the captain of the army and the high priest in favor of the family of David by appointing Benaiah as captain of the army to replace Joab, and Zadok as the high priest to replace Abiathar (v. 35).
Finally, Solomon ended the factors of rebellion by sentencing Shimei to death (vv. 36-46a). Shimei cursed David when he was escaping from his son Absalom's rebellion (2 Sam. 16:5-13). Shimei later repented to David when David settled Absalom's rebellion, and David pardoned him (19:16-23). Solomon dealt with Shimei according to his father David's charge (1 Kings 2:36-38, 8-9). Shimei broke the oath he made to Solomon, and Solomon put him to death (vv. 39-46a).
After all the factors of rebellion had been cleared up, the throne of Solomon was fully established. Thus, the kingdom was established in the hand of Solomon (vv. 46b, 12).