Is it permissible for sisters to distribute their living mother's gold among themselves without the knowledge of their father or brother, and does the sin of not distributing fall upon the sister who refuses to do so?

As long as the mother is alive and has not yet divided the gold, her daughters are not permitted to take or sell it except with her permission, because what was issued from her is not considered a gift. Even if she intended it as a gift, it becomes void due to her life-threatening illness before the gold was taken into possession. The mother must be fair among her children if she wishes to gift the gold, so she should not give to some and deprive others, in accordance with the Hadith: "Be mindful of Allah and be fair among your children."