What is the ruling on the prayer of someone who forgot the takbir (exclaiming "Allahu Akbar") when rising from the second prostration, completed their prayer, then remembered after the taslim (salutation of peace ending the prayer) and performed a third prostration, then performed the prostrations of forgetfulness (sujud al-sahw)? And is the prayer of someone who did not perform the third prostration valid?

The 'takbirat al-intiqal' (transitioning takbirs) are considered sunnah by the majority of scholars, and obligatory by the Hanbalis. If the Imam forgets them, there is nothing upon him according to the majority, but he must perform the prostration of forgetfulness (sujud al-sahw) according to the Hanbalis. If the Imam mistakenly performs a third prostration, and he does so knowingly and intentionally, his prayer is invalidated. However, if he performs it thinking it is a forgotten prostration, his prayer is valid, and he only needs to perform the prostration of forgetfulness. The prayer of the congregants (ma'mumin) is valid even if they unknowingly follow the Imam in the third prostration. As for anyone who follows him knowingly and intentionally, their prayer is invalidated.