What is the ruling regarding a husband greeting his brother's wife, knowing that his wife suffers from extreme jealousy and that the husband refuses to stop doing so?

Shaking hands between a man and a non-mahram woman is not permissible, because it is a cause of fitnah (temptation/strife). It is disliked to greet a young non-mahram woman with Salam, and likewise for her to return it. Some scholars have even ruled that it is forbidden for her to return it. The wife should gently and kindly advise her husband to repent from being lenient in these matters.