Jamieson, Fausset and Brown on Ecclesiastes 4:13

13. The "threefold cord" [Ec 4:12] of social ties suggests the subject of civil government. In this case too, he concludes that kingly power confers no lasting happiness. The "wise" child, though a supposed case of Solomon, answers, in the event foreseen by the Holy Ghost, to Jeroboam, then a poor but valiant youth, once a "servant" of Solomon, and (1Ki 11:26-40) appointed by God through the prophet Ahijah to be heir of the kingdom of the ten tribes about to be rent from Rehoboam. The "old and foolish king" answers to Solomon himself, who had lost his wisdom, when, in defiance of two warnings of God (1Ki 3:14; 9:2-9), he forsook God.

will no more be admonished--knows not yet how to take warning (see Margin) God had by Ahijah already intimated the judgment coming on Solomon (1Ki 11:11-13).