The Ruling on Shaking Hands With One's Brother's Wife

Question :

Is it permissible for a man's brothers to shake hands with his wife, without being alone with her, only in the presence of the sisters and the parents, and usually, it happens during celebrations, such as 'Eid and the like?

Answer:

It is not permissible for a man's brothers, his paternal or maternal uncles, or his cousins, to shake hands with their brothers' wives, or their maternal or paternal uncles' wives. They are like the rest of the unrelated women, because the brother is not a Mahram for his brother's wife and likewise, the paternal uncle is not a Mahram for his brother's son's wife, nor is the maternal uncle a Mahram for his sister's son's wife, nor are the sons of the paternal uncle Mahrams for the wives of their cousins, according to the saying of the Prophet:
"Verily, I do not shake hands with women."

And 'A'ishah, may Allah be pleased with her, said: "By Allah, the hand of the Messenger of Allah never touched the hand of a woman. He did not accept the pledge of allegiance from women except in words."

and because shaking hands with unrelated women might lead to some temptation (Fitnah) over them, like looking at them, or even more so. As for the Mahrams, there is no objection to them shaking their hands, such as the sister, the paternal aunt, the father's wife and the son's wife. And Allah is the Granter of success.

Source:
Ibn Baz
Fatawa Islamiyah, Vol. 5 Pages 111-112