What is the ruling on a wife remaining alone in the house with her husband's brother, who is unmarried and whose wife is not present, while her husband is absent due to his work and his marriage to another woman? Is there any impropriety in their meeting in the shared areas of the house, day and night, especially given that the husband asked her to close her room door?

It is the wife's right that her husband provides her with an independent dwelling along with its amenities. The wife may agree to live with her husband's family, provided that this does not lead to her being alone with a non-mahram (definition) man. This is based on the Prophet, peace and blessings be upon him, who said: "No man is alone with a woman but the third of them is Satan," and his saying: "Beware of entering upon women... the in-laws are death." Therefore, the husband must provide an independent dwelling or prevent her from being alone with his brother.