What are the Nezer and the Edus?
Rashi: [The crown 1 and] the Sefer Torah 2 - "v'Haysah Imo v'Kara Vo Kol Yemei Chayav" (Devarim 17:19).
Rashi citing Avodah Zarah 44a: The crown is the testimony; everyone [from Beis David] who is proper for kingship, the crown fits him. If he is not proper for kingship, the crown does not fit him 3 .
Radak: It is [the crown and] royal clothing, like "v'Adis Edi" (Yechezkel 23:40). The grammatical form of "Edus" is like "Ge'us."
'Rashi' (Divrei ha'Yamim II, 23:11): It is the crown that David took from Rabas Bnei Amon - "va'Yikach David Ateres Malkam" (Divrei ha'Yamim I, 20:2). From then and onwards, all Malchei Yehudah wore it.
Malbim: Really, he needs to write a Sefer Torah after he becomes king; it was given to him at the time [to strengthen his anointment]. It is not clear whether he fulfills the Mitzvah through a Sefer Torah that he commanded to write it while he was a minor.
Radak (from 44a): "Adoniyahu
Who anointed him?
Radak (from Divrei ha'Yamim II, 23:11): Yehoyada and his sons. Malbim - anointment must be via a Navi; Zecharyah ben Yehoyada was a Navi.
We do not anoint a Melech Ben Melech!
Radak, Malbim (from Horiyos 11b): He was anointed due to the opposition of Asalyah, just like Shlomo was anointed due to the opposition of Adoniyahu.
What is the meaning of "Yechi ha'Melech"?
Radak citing Targum Yonasan: The king should succeed.