According to the website: https://www.gotquestions.org/Solomon-Adonijah-Abishag.html
“Solomon rightly saw Adonijah’s desire to marry Abishag as part of his brother’s ongoing attempt to take over the kingdom of Israel.
In those days of royal harems, taking possession of a king’s concubines was a declaration of one’s right to the throne. This had been one of Absalom’s methods when he led a coup against David (2 Samuel 16:22). Since Abishag was considered part of David’s harem, her marriage to Adonijah would have strengthened the usurper’s claim to the throne.”
Another great explanation I found on a Catholic forum. User Rocks_91 says “This Adonijah had previously declared himself king. David wanted his son Solomon to be king, as he had promised his wife that it would be so, and had Solomon anointed and crowned.
When Adonijah made the request to Bathsheba to take one of Solomon’s servants as his wife, of course Solomon must have thought that the guy had some nerve. Solomon saw him as a very dangerous person and a potential threat to his kingdom, and thus had him killed.” https://forums.catholic.com/t/why-did-solomon-have-adonijah-killed/255524 The full conversation and thread can be found here: