Is it permissible for my husband to look at my sister with lust?

The wife's sister is a non-mahram to him. Therefore, it is not permissible for him to be in seclusion with her or to be overly familiar in conversation with her. If there is a need to speak, it should be within the bounds of Quranic etiquette, while also observing Allah's command to lower the gaze, as that is purer for his heart and soul.