Praise be to Allaah.
Yes, that is permissible, because when a man makes a marriage contract with a woman, his father becomes a mahram for her, as does his son from another wife. And her mother becomes a mahram for him, as does her daughter from another husband.
This is called a mahram by marriage.
The evidence that the husband’s father is a mahram for the son’s wife is the verse in which Allaah mentions the women who are mahrams (interpretation of the meaning):
“the wives of your sons who (spring) from your own loins”
So the wife of the son is a mahram for her husband’s father.
The evidence that the husband’s son is a mahram for his father’s wife is the verse in which Allaah says (interpretation of the meaning):
“And marry not women whom your fathers married, except what has already passed”
The evidence that the wife’s mother is a mahram for her daughter’s husband is the verse in which Allaah mentions the women who are mahrams (interpretation of the meaning):
“…your wives’ mothers”
These three (the husband’s father, his son and the wife’s mother) become mahrams as soon as the marriage contract is done, and that is not subject to the condition that the marriage be consummated.
With regard to the wife’s daughter, she does not become a mahram for her mother’s husband unless he has consummated the marriage with her mother, because Allaah says (interpretation of the meaning):
“ and your stepdaughters under your guardianship, born of your wives to whom you have gone in — but there is no sin on you if you have not gone in them (to marry their daughters),”
The stepdaughter (rabeebah) is the daughter of one's wife.
See al-Mughni, 9/514, 524.
In conclusion: the husband’s father is a mahram for his son’s wife, so he may shake hands with her and be alone with her and travel with her. See questions no : 5538 , 20750
And Allaah knows best.