Is it permissible for the husband’s brother to take his brother’s wife to the doctor?

It is not permissible for the wife to travel in a car alone
with her husband’s brother, because that comes under the heading of khulwah
(being alone with a member of the opposite sex) which the Messenger
(peace and blessings of Allaah be upon him) warned against when he said:
“Beware of entering upon women.” They said: “O Messenger of Allaah, what
about the in-law?” He said: “The in-law is death.” Narrated by al-Bukhaari
and Muslim. 

What do you understand from these words? Is it forbidden or
allowed? Undoubtedly what is meant is that it is forbidden and not allowed.
So it is not permissible for a man to be alone with his brother’s wife,
either in a car or in the house. Worse than that is what some people do when
a guest comes when the man is at work, and there is no one in the house
except his wife, then she opens the door to him and he goes in and waits for
the owner of the house to come home. The point is that it is not permissible
for any woman to be alone with any man (i.e., non mahrams) even if he is one
of her husband’s relatives or one of her relatives or neighbours, unless she
has a mahram with her, whether that is in her home town or when travelling.
It should also be noted that it is haraam for her to travel even if there is
no khulwah, if she does not have a mahram with her, because of the hadeeth
of Ibn ‘Abbaas (may Allaah be pleased with him) which is narrated in
al-Saheehayn. He said: I heard the Prophet (peace and blessings of
Allaah be upon him) delivered a khutbah and said: “No man should be alone
with a woman unless there is a mahram with her, and no woman should travel
without a mahram.”