Is it permissible to shake hands with one's brother's wife or to greet her without shaking hands?

The wife of the brother is a non-mahram (definition), so it is absolutely impermissible to shake her hand, based on the Prophet's (peace be upon him) saying: "Indeed, I do not shake hands with women," and based on Aisha's (may Allah be pleased with her) saying: "The hand of the Messenger of Allah (peace be upon him) never touched the hand of a woman; he would only take their oath of allegiance verbally." There is no difference between shaking hands with a barrier or without one, to block all pretexts leading to evil. As for greeting verbally without shaking hands, it is permissible if there is no fear of temptation and no private seclusion, as was reported from Umm Darda and Aisha (may Allah be pleased with them both), and as was the case with women during the time of the Prophet (peace be upon him).