So I'm going to be the devil/Adonijah's advocate here and see where it takes me.
The reason why he was not executed here was because Solomon showed mercy but it was conditional.
Or so Solomon said, right? From which side in this disagreement do we get this account? Solomon's or Adonijah's? And why isn't Joab treated to the same condition when he grabs the horns in the sanctuary? Seems rather arbitrary on Solomon's part, unless you consider that he couldn't kill Adonijah right off the bat since Adonijah seems to have had some significant support. Doing so would have been a stupid move politically, especially if Solomon, rather than Adonijah, was the usurper.
Adonijah broke the agreement and was conspiring against Solomon again so the blood was on his own head.
Did Adonijah conspire against Solomon or did Solomon actually conspire against Adonijah? Maybe Adonijah was the legitimate successor, as would be expected, but Bathsheeba took advantage of David's weakened mental state?
In normal circumstances, Solomon would not have inherited the throne because he was not the oldest son of King David. Adonijah wanted to rise to power so he went about doing this by trying to make himself look like the legitimate successor even though Solomon was given the throne. He wanted to take Abishag as a wife because of her close connection to King David to further strengthen his case. Solomon knew what he was doing so this ended the agreement they came to.
Maybe Solomon's mother usurped the legitimate successor, Adonijah?
After all, David's two closest advisors - his general, Joab and his high priest, Abiathar - supported Adonijah.
1 Kings 1:7 He collaborated with Joab son of Zeruiah and with Abiathar the priest, and they supported him.
These aren't just two random guys. Joab and Abiathar had been with David from the beginning of David's rise to power. Joab was David's general and Abiathar was David's high priest, who David trusted enough to be one of his inside men/spies when David was forced to flee from Absolom.
2 Sam 15:27 The king said to Zadok the priest, Are you a seer? Go back to the city in peace! Your son Ahimaaz and Abiathars son Jonathan may go with you and Abiathar. 15:28 Look, I will be waiting at the fords of the desert until word from you reaches me. 15:29 So Zadok and Abiathar took the ark of God back to Jerusalem and remained there.
Somehow, two of David's closest compatriots supported Adonijah and didn't seem to think anything about it. They didn't seem to think they were usurping Solomon.
Adonijah holds his coronation right outside Jerusalem in En Rogel. He doesn't appear to know he's doing anything wrong. 1 Kn 2 even says that he recognized that (a) he was the legitimate successor and (b) he had the people's support. He manages to acquire the royal trappings of chariots and a royal guard. He invites David's other sons (his brothers, except Solomon), royal servants, the men of Judah, and holds a feast full of sacrifices. How is he able to fool all of these people, and not just ordinary people who might not know better, into thinking he was the legitimate successor? Probably because he actually was the legitimate successor. The text even tells us that David doesn't make an effort to fix Adonijah, which implies that David himself seems to have seen Adonijah as the legitimate successor. Solomon wasn't anywhere near the front of the line for succession.
Solomon on the other hand only manages to get David's blessing as successor after David's mental state is clearly in question. And Bathsheba approaches David only after Adonijah is starting to claim the throne, seemingly at the suggestion of Nathan.
The text says "all the people" declared "long live king Solomon" and "all the people" followed Solomon celebrating. But can this be the case? Some of the most important people - the high priest, David's general, royal servants, and David's other sons clearly aren't there. They are busy over at Adonijah's coronation.
This smells like a conspiracy between Bathsheba, Nathan, and Benaiah to usurp the legitimate successor, Adonijah. And what do you know? Before too long, Joab and Adonijah will both be dead with clear implication that Bathsheba is involved in the death of at least Adonijah and the explicit statement that Benaiah kills Joab.
Solomon showed Adonijah tremendous mercy by not sentencing him to death immediately following his coronation, knowing that Adonijah had every intent to kill him and his mother had he become King.
Do we see evidence that Adonijah intended to kill Solomon? I know that's what Bathsheba claims, but is that really he case? Why believe Bathsheba?
The only evidence of anyone intending to kill anyone else is when Solomon and Bathsheba actually do kill Adonijah and Joab.
When Adonijah had the gall to ask for Abishag when he should have lived very humble and quietly,
So he should have just shut up and color. What if he was the legitimate successor?
Solomon knew without a doubt that Adonijah could not control his lust and would stop at nothing to usurp the throne as long as he remained alive. His death was justified.
Adonijah's attempt at marriage was clearly a political move, not a lustful one. Besides, it appears to be a rather suspiciously clumsy political move. And why think that he would continue to try to "usurp" the throne as long as he lived?
It's a little strange how nonchalantly some people are ready to justify the execution of Adonijah. Solomon was good with it, so I guess we must be good with it too?
[/end bad guy position]