After Abner's death, 2 brothers (who had been under Abner's command) went and killed their own king, lshbosheth. They killed him in his own home; in his own bed. And, mistakenly they assumed that David would be thrilled that his enemy was dead. So, they brought him lshbosheth’s head. But, they were dead wrong. David had those 2 men executed just like he had done to the Amalekite who had formerly brought him Saul's crown.
In the end, the most important thing (as far as the development of Israel's history is concerned) is that after this episode, there was only one king in Israel. And, David was not nervous, insecure, jealous nor vindictive as a young king. He had God's distinct hand of blessing upon him as a leader. He had the discernment, discipline and direction that are necessary for good leadership. His reign was certainly going to be a step up from that of his predecessor, Saul. And, the difference maker is seen in 1 primary area: faith. David believed God... he was a man of faith, whereas Saul is remembered for his doubts and his disobedience.