Or HaChaim on Genesis 14:12/אור החיים על בראשית י״ד:י״ב

1 ויקחו את לוט, They took Lot, etc. The reason the Torah adds that Lot was Abraham's nephew and that he lived in Sodom, something we have known for a long time, is to tell us that these kings were very well aware of who Lot was. This did not deter them from taking Lot captive. The Torah stresses that these kings deliberately ignored the fact that they were provoking a man of international repute when they took his nephew into captivity. None of this would have happened to Lot had he not chosen to live in Sodom. The matter can be compared to Israel capturing a substantial part of Moab though G'd had forbidden Israel to provoke war with Moab or to attack it (Deut. 2,9). Once the Emorite king Sichon had defeated Moab in war, the lands captured by Sichon were no longer considered as Moabite territory, and Israel took possession of them. Here too, once Lot had chosen to throw in his lot with the Sodomites he was no longer considered as part of Abraham's entourage, and was not entitled to the protection Abraham could have afforded him. This also explains why Abraham did not try to secure Lot's release by peaceful means. He knew that the 4 kings knew very well whom they had taken prisoner. If Lot's relationship with Abraham had not stopped them, there was no point for Abraham to warn those kings before attacking them. Lot's capture was a hostile act against Abraham.

2Should you argue that the four kings had merely reacted against the insubordination of the five kings associated with the king of Sodom, and that Lot was their legitimate booty, as per the comparison we have drawn with Sichon and the Moabites, this is not an accurate comparison after all. That comparison was valid when there existed neither a state of peace nor a state of war between two parties. Abraham and the four kings were at peace, and the four kings should have warned Lot to leave Sodom before making war just as did King Saul with the Kenites before he attacked the Amalekites (Samuel I 15,6).