Deuteronomy 24:1

When a man taketh a wife Deuteronomy 22:13.

then it shall be … that he shall write her, etc.] Rather, and it come to pass … that he write her, etc. The apodosis does not commence here but in Deuteronomy 24:4.

some unseemly thing As in Deuteronomy 23:14 (15), the nakedness of a thing, something indecent or repulsive, LXX ἄσχημον πρᾶγμα. The expression is so indefinite that it gave rise to controversy in the Rabbinic schools; that of Shammai understanding by it unchastity, that of Hillel any physical blemish or other, even the most trivial, cause of dislike. It cannot be adultery for this was punished by death. The words suggest some immodest exposure or failure in proper womanly reserve.

bill of divorcement Lit. of separation. Bill, Heb. sepher, used of any missive (e.g. 2 Samuel 11:14 f.) or legal deed (Jeremiah 32:11), as well as book, LXX βιβλίον. Something in legal form, and possibly procurable only from some public authority. Yet, notice, there is no mention of elders here as in the procedure in Deuteronomy 22:13-21. The later Jews called such a document geṭ, and the procedure in connection with it is prescribed in the Mishna, -Giṭṭin."

and give it … her … and send her Two further formal steps of personal service of the deed, and the husband's own solemn dismissal. So his responsibility in the matter is not weakened.