In Divrei HaYamim I:24 verses 3 and 6, we read the following:
"And David distributed them, both Zadok of the sons of Eleazar, and Ahimelech of the sons of Ithamar, according to their offices in their service."
And Shemaiah the son of Nethaneel the scribe, one of the Levites, wrote them before the king, and the princes, and Zadok the priest, and Ahimelech the son of Abiathar, and before the chief of the fathers of the priests and Levites: one principal household being taken for Eleazar, and one taken for Ithamar.
This is to make the house of Zadok into descendents of Aharon, thus making Zadok kosher. Before, Zadok is simply called the Cohen (Melachim I:4), or the Cohen of the Bnei Levi (Ezechiel 40,43,44,48). Note that it is not the first time that I found Divrei HaYamim to do this contemptible thing. This is what I meant by "those, in the religious politics of the time," who committed "utter falsehood," and who would have both groups come from Aharon, one from his son Elazar and one from Itamar his son.
It is an attempt to make reality subject to "the truth", but the truth is that the truth different. J1 and J2 cohanim are, in reality, incompatable, but their being the nucleus of Am Yisrael, that puts us rightly in Middle East.
Now, there is a further background to "the truth" of Divrei HaYamim. B'Ezrat HaShem, this "truth" and its background will be the subject of subsequent posts.