Is it permissible to say, "O Allah, this is on behalf of our Prophet Muhammad" when offering the sacrifice (Udhiyah), and if not, what is the correct thing to say?

The fundamental principle in acts of worship is tawqif (restriction to what is prescribed), and there is no narration from the Salaf (early generations) regarding offering sacrifice on behalf of the Prophet, peace and blessings be upon him, except for what was narrated from Ali ibn Abi Talib, may Allah be pleased with him, which Al-Albani deemed weak. Therefore, sacrificing on behalf of the Prophet, peace and blessings be upon him, either means that you did not sacrifice for yourself, which is a grave error, or you intended to include him with you, which is not valid because the sacrifice is only sufficient for the household. Alternatively, it means you slaughtered another sheep for him, and this falls under sacrificing on behalf of the deceased, a matter on which scholars have differed. Those who prohibit it argue that it was not known among the Salaf, while those who permit it contend that the Prophet, peace and blessings be upon him, sacrificed for those among his Ummah who did not sacrifice. However, it was not reported that the Rightly Guided Caliphs and those who came after them did so. When offering the sacrifice, one should say: "Bismillah, Allahu Akbar. O Allah, accept it from me and my family," or "from so-and-so and the family of so-and-so."