What is the ruling on someone who, after rising from rukūʿ, inadvertently said "Allāhu akbar" and bowed again, then remembered and proceeded to prostrate, and then said "Allāhu akbar" again to transition from bowing to prostration? And does the prayer become void due to the extra takbīr? And what is the ruling on someone who inadvertently performed the prostration of forgetfulness (sujūd al-sahw) before the salām instead of after the salām?

The takbir (proclamation of 'Allahu Akbar') for transitioning between prayer postures is obligatory according to some scholars. If it does not occur in its proper place, it is not sufficient. The takbir for lowering, rising, and standing should begin with the commencement of the transition and end with its completion. If the worshipper completes it within a portion of the transition, it is sufficient. However, if they begin it before or complete it after, so that some of it falls outside the transition, it is as if they have omitted it.

Therefore, the takbir for transitioning into prostration is the prescribed one. If the worshipper were to only rely on the additional takbir for bowing, they would have performed the takbir of transition out of its proper place. In this case, it is Sunnah for the worshipper to perform the prostration of forgetfulness (sujood as-sahw) after the salam, according to the preponderant opinion. However, if they perform the prostration of forgetfulness before the salam due to an oversight, there is no harm in it, and their prayer is valid in any case.