What is the ruling on a husband's dealings with his brother's wife, especially given that she is immodestly dressed (mutabarrijah), and she congratulates him on his birthdays and corresponds with him, and he praises her, knowing that he meets her in his family's home?

Extreme caution is required in a man's relationship with his brother's wife, due to the potential for fitna (strife/temptation) therein. The Prophet, peace and blessings be upon him, warned against the husband's relatives entering upon his wife, likening the hamu (male in-law) to death. The hamu in the hadith refers to the husband's relatives other than his fathers and sons, such as the brother and the nephew, as people tend to be lax in this regard, and they are more deserving of being prevented than a stranger. Therefore, it is not permissible for your husband to be overly casual in his relationship with his brother's wife, nor to speak with her about private matters, especially if she is immodestly dressed. If they are gathered for an occasion, a place should be designated for women separate from men. If there is a necessity to speak, it should be limited to the need, without khalwa (private seclusion) or suspicion, while adhering to Islamic manners such as lowering the gaze.