Was Avishag permitted to him?
Radak: There was no Isur [of Eshes Av], for David never had Bi'ah with her 1 . However, a commoner may not use a king's scepter (she is no less than David's scepter)!
Had David had Bi'ah with her (without Kidushin, for he could not marry another wife), she would have been forbidden to David's sons according to R. Yehudah due to "Kenaf Aviv" (Devarim 23:1). (PF)
Why did he request Avishag? Was he not concerned lest Shlomo think that he wants her to establish himself as king (only a king may marry a king's widow)!
Mahari Kara (27): Hashem put this in his heart, in order to fulfill "he will pay four-fold for the lamb" (Shmuel II, 12:6. David lost four children - Bas Sheva's first baby, Amnon, Avshalom and Adoniyahu 1 .)
Radak: Indeed, he requested her in order to try to establish himself as king! (He hoped that Shlomo would not realize. Indeed,) Bas Sheva did not realize his intent.
Malbim (3): He desired her so much that he was willing to risk his life for her 2 . From when he saw her in his father's house he put his mind on her. Desire burned in his heart to reign immediately and take her for a wife. David would not withhold her from him, since he did not have Bi'ah with her; she merely served him. David can find other beautiful girls to serve him!
Even though Adoniyahu died after David, it was considered part of David's punishment. (Some say that Kil'av (Avigayil's son) died in David's lifetime (refer to 1:5:2:2 and the note there). Why is this not considered part of David's punishment? Kil'av was one of the four who never sinned! (Shabbos 55b - PF) Targum Yonasan in Shmuel II says that David was punished with afflictions via four children - the baby, Amnon, Tamar and Avshalom.
If this was his only intent, he should have offered to admit in front of the Sanhedrin and army officers 'even if I marry Avishag, anyone who says that I have any claim to kingship, he is Chayav Misah for Mored b'Malchus Shlomo!' Would this not dispel concern that he intends to usurp kingship?! (PF)