What is the way to deal with a married woman who cheated on her husband and refuses to return to him, and is seeking a divorce to marry another, while her family fears scandal and her brother might kill her?

Adultery is a grave sin, and the adultery of a married woman is even more heinous. She must be reminded to repent to Allah with its conditions. If she repents, it is good for her husband to keep her and live with her kindly. She does not need to observe a waiting period (istibra’) if she is already married.

It is not permissible for her to seek a divorce without a legitimate Shar'i justification. Her desire to marry another man does not justify her seeking a divorce.

If she does not repent, it is better for her husband to separate from her. He may also ostracize her if there is a strong hope that such ostracization will achieve a significant benefit; otherwise, not.