John Gill Commentary 1 Kings 1:13

"Go and get thee in unto king David, and say unto him, Didst not thou, my lord, O king, swear unto thy handmaid, saying, Assuredly Solomon thy son shall reign after me, and he shall sit upon my throne? why then doth Adonijah reign?" — 1 Kings 1:13 (ASV)

Go and get yourself in to King David
That is, go into the chamber where the king lay, at once, without any ceremony:

and say to him, did you not, my lord, O king, swear to your
handmaid, saying, assuredly Solomon your son shall reign after me, and
he shall sit upon my throne ?
though no mention is elsewhere made of such an oath, there undoubtedly was one, of which Nathan has knowledge, either from David or Bathsheba, or from them both, or might be present himself at the making of it; for not only Bathsheba affirms it, (1 Kings 1:17) ; but David owns it and confirms it, (1 Kings 1:30) ;

why then does Adonijah reign ?
surely it cannot be with the king's knowledge and consent, so manifestly contrary to his promise and oath.