What is the ruling on a girl intentionally showing her hair to her brother with the aim of demonstrating that she is more beautiful than his wife?

The basic principle is that a woman is permitted to uncover for her mahrams—such as her father and brother—what is customary to be uncovered, and this includes her hair. However, it is impermissible for her to uncover if it leads to fitna (temptation or discord) from her mahrams, or if she has an illicit intention, such as provoking her brother's wife, or inciting hatred towards his wife because of her lack of beauty, or making him feel that he was deceived into marrying her. A woman will be punished for such actions and corrupt intentions, as the Prophet (peace be upon him) said: "Indeed, actions are by intentions," and he also said: "He is not of us who incites a woman against her husband."