'King of Israel' in the Bible

After whom did the king of Israel go out? After whom are you pursuing? After a dead dog? After one flea?

And so then, do not let my blood fall to the ground {away from} the presence of Yahweh, for the king of Israel has gone out to seek a single flea, as one hunts a partridge in the mountains."

Then the commanders of [the] Philistines said, "What [are] these Hebrews [doing here]?" And Achish said to the commanders of [the] Philistines, "Is this not David, the servant of Saul, the king of Israel, who has been with me {for days and years}? I have not found anything [threatening] in him from the day of his desertion until this day!"