Carl Friedrich Keil and Franz Delitzsch Old Testament Commentary

After the anointing they were to conduct Solomon up to Zion again; Solomon was then to ascend the throne, as David was about to appoint him prince over Israel and Judah in his own stead. Both the anointing and the appointment of Solomon as prince over the whole of the covenant nation were necessary, because the succession to the throne had been rendered doubtful through Adonijah's attempt, and the aged king was still alive. In cases where there was no question, and the son followed the father after his death, the unanimous opinion of the Rabbins is, that there was no anointing at all. Israel and Judah are mentioned, because David had been the first to unite all the tribes under his sceptre, and after the death of Solomon Israel fell away from the house of David.