What is the ruling of Islamic law regarding a brother who instills hatred in his mother's heart against his (other) brother, and will the mother's supplication against her son be answered in this case, and who bears the burden of sin?

The mother's right is great, and honoring her is the most obligatory of duties. Disobeying her is one of the greatest sins, especially during her weakness and illness. Therefore, it is not permissible to cut ties with her, no matter how much she wrongs, and one must consistently visit her, honor her, and satisfy her in ways that do not violate Islamic law. The wife is not obligated to visit the husband's mother. Respond to the mother's wrongdoing with kindness so that animosity may cease and affection may be found. If you honor your mother and treat her well, her anger or her supplication against you unjustly will not harm you, for a supplication for sin is not answered. Do not accuse your brother of causing discord between you and your mother without proof, for the default state of a Muslim is innocence and good assumption. If you are certain of his causing discord, then it is a great evil detested by Allah, and one who causes discord between people is considered to be shaving away religion and reward. Your brother has the right to connection, advice, and supplication for him, no matter his condition.