Then Bathsheba bowed with [her] face to the earth, and did
reverence to the king
Thereby expressing her veneration of him, and thankfulness to him for his favour to her and her son, in fulfilling his promise and oath:
and said, let my lord King David live for ever;
which though a common form of salutation of kings, not only in Israel, but in other nations, is not to be considered as a mere compliment, but as expressing the real desires and affection of her heart to the king; signifying hereby that her solicitations on the behalf of her son did not arise from any desire of the king's death; she heartily wished him health to live long and easy; and all her request was, that Solomon her son might succeed him, whenever it pleased God to remove him; or seeing he was now a dying man as it were, her prayer was that his soul might live for ever in happiness in the world to come; so Kimchi interprets it.