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 husbandâs father is a mahram for the sonâs wife is the verse in which Allaah mentions the women who are mahrams (interpretation of the meaning):
âthe wives of your sons who (spring) from your own loinsâ
So the wife of the son is a mahram for her husbandâs father.
The evidence that the husbandâs son is a mahram for his fatherâs wife is the verse in which Allaah says (interpretation of the meaning):
âAnd marry not women whom your fathers married, except what has already passedâ
The evidence that the wifeâs mother is a mahram for her daughterâs husband is the verse in which Allaah mentions the women who are mahrams (interpretation of the meaning):
ââ¦your wivesâ mothersâ
These three (the husbandâs father, his son and the wifeâs 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 wifeâs daughter, she does not become a mahram for her motherâs husband unless he has consummated the marriage with her mother, because Allaah says (interpretation of the meaning):
â and your stepdaughters under your guardianship, born of your wives to whom you have gone in â but there is no sin on you if you have not gone in them (to marry their daughters),â
The stepdaughter (rabeebah) is the daughter of one's wife.
See al-Mughni, 9/514, 524.
In conclusion: the husbandâs father is a mahram for his sonâs 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.