There is always a king. The rightful king is always out there. He is the noblest man of his generation, and (by definition) there is always such a man. The only question is whether he is known, recognized and honoured as such. Where he is not, all men worry whether they might themselves be the rightful king; and, thinking they might be, feel resentment that their dignity is not properly recognized. A fortiori, they resent anyone who lords it over them. In such circumstances is individual liberty most jealously, zealously guarded. In such circumstances, it must be.
In the absence of an identified king, all authority is sapped, all the way down the hierarchy – for the same dynamic is at work for every level of the patriarchy. Such is chronic kinglessness, in which everyone has some power and no one feels himself powerful. Kinglessness engenders anomie, for all laws of the kingless state are unlegitimated by any royal authority. All laws and rulings are then under radical question, their power of moral suasion uncertain, as being indefinite, up for grabs; so that obedience and lawfulness become a matter of mere expedience. Chaos ensues; no one is or feels safe, or reliably correct.
Authority is then never quite settled, and political strife is constant. Where the king is known, and honored, there is no political strife, because his edicts are Law, and are everywhere accepted as such. Political strife arises only when there is some question about who is the rightful king. It is a war more or less violent to see who will rule. Where no rule is royal, ergo sacrosanct, there is no security of rule, or therefore of law.
In any group of men under stress the noblest among them will soon become apparent to all. All will want to be his friends and allies, and will want to be like him. Ditto for any group of nobles.
No formal procedure is needed for royalty to emerge in this way.