Albert Barnes Commentary 1 Kings 1:12

"Now therefore come, let me, I pray thee, give thee counsel, that thou mayest save thine own life, and the life of thy son Solomon." — 1 Kings 1:12 (ASV)

It would have been in accordance with general Eastern custom for Solomon to be put to death if Adonijah had succeeded in his attempt. But to have executed his mother as well would have been an unusual severity. Still, such cases sometimes occurred: Cassander put to death Roxana, the widow of Alexander the Great, at the same time as her son, the young Alexander.