What is the Sharia ruling concerning a mother whose adultery was proven after 37 years, and that two of my siblings are not from my deceased father, knowing that she constantly criticizes and slanders people's honor and causes us problems, and she does not remember the real father of her children? And how do I deal with them in the future, given the fear of Allah's punishment and the inability to forgive her or remain silent about her actions?

Accusing your siblings of immorality and slandering their lineage without clear proof is a grave evil. They are related to their father, who is your mother's husband, and every child born on the marital bed is attributed to the husband. Therefore, you must maintain kinship ties with them, and severing ties with them is forbidden. If your mother committed adultery, she must repent sincerely and conceal her sin. Her right upon you is great and is not nullified by her sin or wrongdoing.