Were "al-Sajjad" and "al-Baqir" among the sons of Ali ibn Abi Talib, and was "al-Kadhim" among the sons of Ja'far ibn Abi Talib?

In a previous fatwa, the wives and children of Ali ibn Abi Talib, may Allah be pleased with him, were clarified. As-Sajjad is Muhammad ibn Talha, nicknamed As-Sajjad due to his devotion. He was born during the lifetime of the Prophet, peace and blessings be upon him, and was killed young on the Day of the Camel. His mother is Hamnah bint Jahsh.

As for Abu Ja'far al-Baqir, he is Muhammad ibn Ali ibn al-Husayn ibn Ali ibn Abi Talib. He is the son of Ali's grandson, may Allah be pleased with him, and not his direct son.

As for al-Kazim, he is Musa ibn Ja'far ibn Muhammad ibn Ali ibn al-Husayn ibn Ali. He is the son of Abu Ja'far al-Baqir and the father of Ali ibn Musa al-Rida. Al-Dhahabi described him as the Imam, the exemplary leader, the master, Abu al-Hasan al-Alawi, from Medina, who resided in Baghdad. There is no one known among the children of Ja'far ibn Abi Talib by the name of al-Kazim; rather, he is the son of Ja'far ibn Muhammad.