John Gill Commentary 1 Kings 1:17

"And she said unto him, My lord, thou swarest by Jehovah thy God unto thy handmaid, [saying], Assuredly Solomon thy son shall reign after me, and he shall sit upon my throne." — 1 Kings 1:17 (ASV)

And she said unto him, my lord, you swore by the Lord your
God unto your handmaid
Which was a very solemn oath, and binding, and which she puts David in mind of, knowing that so conscientious a man as he was would religiously observe it:

[saying], assuredly Solomon your son shall reign after me, and shall sit
upon my throne ;
be his successor in it, and established on it.