Is it permissible to take the mother to live away from the father who only prays on Fridays, slanders people, and accuses his wife of treachery, knowing that he is elderly and suffers from health problems, and that the mother is diligent in prayer, given the father's unwillingness to listen to anyone and the difficulty of changing his nature?

The basic principle is that a mother is not permitted to leave her husband's home without his permission, unless he is abusing her with severe beatings, in which case she is then permitted to leave him.

If the mother is harmed by remaining with the father, she is permitted to separate from him through divorce or khulʿ (a type of divorce initiated by the wife), and the children are allowed to help her with this.

Under no circumstances are the children permitted to sever ties with their father or treat him harshly; for the father's right is great, and his right to dutifulness and companionship with kindness does not fall, no matter how much he wrongs, just as Allah commanded companionship with kindness even for polytheistic parents.

The children must be dutiful to their mother and father without harming either of them or neglecting their rights. Part of being dutiful to the father is enjoining him to do good and forbidding him from evil gently, striving to reform him, and helping him to repent and adhere to obedience to Allah, especially prayer, which is the pillar of Islam.

Seeking help through patience and prayer aids in all matters. Patience, in all its forms, is a great help, as is prayer, which forbids indecency and evil.