Is it obligatory to repeat the prayers that were performed with the Abrahamic prayer mentioned in the first Tashahhud by mistake, and then followed by the prostration of forgetfulness (sujud al-sahw), believing that the addition of a pillar is compensated for by the prostration of forgetfulness?

Praying upon the Prophet (peace and blessings be upon him) in the first Tashahhud is neither a pillar nor an obligation; in fact, it is not even recommended according to many scholars, and this is the more preponderant view. If a Muslim intentionally prays upon the Prophet in the first Tashahhud, their prayer is valid. The prostration of forgetfulness (sujud al-sahw) is not prescribed for someone who does this out of forgetfulness, and even if its recommendation is mentioned, it is not obligatory. As for someone who does this out of ignorance, their prayer is not invalidated.