What is the ruling on the action of an Imam who led people in prayer for one rak'ah, then remembered he was not in a state of ritual purity (wudu'), so he قطع the prayer and appointed one of the congregants to lead, who then continued the prayer building upon the Imam's prayer, after which the Imam went, performed wudu', and repeated the prayer, knowing that he adheres to the Shafi'i madhhab?

What the Imam did by withdrawing from the prayer and appointing one of the congregants as his successor, and the successor building upon the Imam's prayer, is a correct action. This is the view of the majority of scholars, and it has been established by the Prophet's (peace be upon him) tradition when he appointed Abu Bakr to lead the prayer. It is also supported by the action of Umar ibn al-Khattab (may Allah be pleased with him) when he was stabbed and put forward Abd al-Rahman ibn Awf to lead the people in prayer. If the Imam does not appoint a successor, then the congregants have the choice between putting forward one of them or completing the prayer individually.