What is the ruling regarding a man telling his wife: "Go stay with your family" (literally, "Flee to your family's house"), while he later forgets about it and is unaware of his intention at that time, and is this considered a threat or an intention of divorce, given his subsequent rejection of the idea of divorcing her?

The phrasing mentioned is one of the allusions to divorce (kinayat al-talaq), and for divorce by allusion, intention is a prerequisite. As long as you doubt your intention, the default is the absence of intention, and thus the wife is not divorced, because doubt regarding divorce does not negate the certainty of the marriage bond's continuation. Ibn Qudamah said: "Whoever doubts his divorce is not bound by its ruling." The wife is advised to avoid doing what the divorce was made conditional upon, as a precautionary measure.