Is it permissible for me to leave the house while my brother's wife does not adhere to the Islamic dress code and does not cover her head inside the house?

It is permissible for a man to live with his brother and his wife if he has a separate room with its own entrance, exit, and bathroom. However, he is not permitted to be alone with his brother's wife. He must lower his gaze from her if he sees her with her head uncovered, just as the brother must oblige his wife to observe the prescribed Islamic hijab.