How did the Ja'fari school of thought originate, and is it considered a fifth school alongside the four existing ones?

Ja'far al-Sadiq was one of the imams of fiqh (Islamic jurisprudence). Scholars such as Abu Hanifa, Ibn Hibban, and Al-Dhahabi praised him. However, his fiqh was not fully codified like the four madhhabs (schools of thought). The adoption of his madhhab did not become a continuous tradition among the Sunnis; rather, some innovators (mubtadi'ah) falsely attributed themselves to his madhhab, introducing into it what was not from him, and lied about him. He himself disavowed anyone who claimed him to be an infallible imam, or who disavowed Abu Bakr and Umar. What is attributed to him from the words of these innovators is rejected because it contradicts the sacred texts and because it was not transmitted by reliable narrators. To understand the current Ja'fari madhhab and its history, one can refer to the book "Ghara'ib Fiqhiyyah 'inda al-Shi'ah al-Imamiyyah" (Peculiar Jurisprudence among the Imami Shi'a) by Mahmud Shukri al-Alusi.