Is it permissible for the one offering the sacrifice to keep one of the four sacrifices for his household, knowing that he did not name any of them at the time of purchase, and one of the sacrifices was on behalf of his brother through an代理 (agency)?

Scholars have differed regarding the designation of an Udhiyah (sacrificial animal) merely by intention. The preponderant opinion is that it is not designated except by explicit verbal declaration, such as saying: "This is a Hadiy (offering)," or "an Udhiyah," or "for Allah," or by signaling (ish'ar) or garlanding (taqleed) with the intention. It is not designated by mere intention at the time of purchase. This is the view of the majority of scholars. You are permitted to keep what you have not verbally designated.