What is the authenticity of the two traditions mentioned in Al-Shafi'i's book "Al-Umm," concerning Ali (may Allah be pleased with him) saying "Peace be upon you, peace be upon you" in the prayer?

The intent behind "these two narrations" in the question is the ruling on saying "Salamun Alaykum" (سلام عليكم) and "As-Salamu Alaykum" (السلام عليكم) in the salutation of prayer.

The two narrations from Ali (may Allah be pleased with him) are authentic, but they are mawquf (stopped at him) and are supported by what Ibn Hibban narrated from Abu Humayd As-Sa'idi. The well-known wording from the Prophet (peace be upon him) is "As-Salamu Alaykum" (السلام عليكم) with the definite article. The Malikis and Hanbalis believe it is obligatory to use the definite article with salam, and their evidence is the Prophet's consistent use of this wording and his saying: "Pray as you have seen me pray." The Shafi'is hold that "Salamun Alaykum" (سلام عليكم) is sufficient, though "As-Salamu Alaykum" (السلام عليكم) is preferable. Imam Shafi'i stated in Al-Umm that the minimum for the salutation is "As-Salamu Alaykum" (السلام عليكم).