Is what the Imam did correct or not, by performing a complete rak'ah after having prostrated once in the first rak'ah, and being alerted to this after the prayer?

The Imam's prayer is valid, but he must perform the prostration of forgetfulness (sujud al-sahw) if he inadvertently omitted a prostration and did not return to it because he did not understand the glorification (tasbih) of the congregants, and then immediately performed an extra rak'ah after the salutation (taslim). As for the congregants, they must alert him. If he does not return, they must perform the prostration he missed and complete their prayer with him. They should not follow him in making up the rak'ah after they have completed their prayer. However, if the Imam realizes the mistake before the salutation and stands up for an extra rak'ah, and the congregants follow him knowingly, their prayer becomes invalid due to their deliberate addition.