What is the ruling on the prayer of someone who, out of forgetfulness, completed the Abrahamic prayers in the first sitting, then forgot to recite Al-Fatiha in the third rak'ah of a congregational prayer?

If the error in reciting the Tashahhud is due to omitting the salutation to the righteous servants of Allah or omitting the testimony that Muhammad is the Messenger of Allah, this does not invalidate the prayer. This is because the first Tashahhud is a Sunnah according to most scholars, and the Imam bears this for the ma'mum (follower). In this Tashahhud, one can restrict it to its wording without the Salat al-Ibrahimiya (Ibrahimic prayer), and if one recites it, there is no harm. If the ma'mum forgets to recite Al-Fatiha behind the Imam, his prayer is valid according to some scholars, but repeating it as a precaution is better.