Is it permissible to go out for recreation with the husband's brother, who is married to a foreign non-Muslim woman, knowing that she does not adhere to Islamic dress, and would that be considered participation in sin when men look at her?

The husband must obligate his non-Muslim wife to wear the hijab, to preserve her chastity. It is also the duty of a Muslim to prevent his Christian wife from displaying her adornments.

As for going out with her without Islamic attire, if it is for the purpose of inviting her to Islam and winning her heart, and the Muslim (man) does not fear being influenced by her, then there is no harm. She bears no sin for men looking at her.

Anything other than that, one should avoid going out with her, as it implies contentment with her actions, and because a bad companion can influence his companion, as the Prophet, peace and blessings be upon him, said: "The example of a good companion and a bad companion is like that of the perfume seller and the blacksmith."