A dictionary of Confusable Phrases: More Than 10,000 Idioms and Collocations

Note: 
Neither expression correlates in meaning with the phrase
under a cloud
— suspected of smth. reprehensible: He leaves the
service, possibly under a cloud…. The inference in the club
think tank was that the cloud was sexual.
under one’s arm • under one’s hand
under one’s arm —held between the arm and the body:
Alfredo wandered about the snow-covered streets with the
manuscript of his last tragedy under his arm.
under one’s hand —(of a document) authenticated by a per-
son’s signature: Lady Bellasis gained so much on the duke,
that he gave her a promise under his hand to marry her.
Note: 
The expression does not correlate in meaning with the
compound underhand—(also: underhanded) 
1
. done in a secret
or stealthy manner: Their influence must have been exercised
in a purely underhand way. 2. not legal; corrupt or marked by
deception: We have found that banks are employing increas-
ingly underhand methods to avoid their responsibility.
under the arm • under-arm
under the arm —
1
. held between the arm and the body: The
hat could be worn or carried under the arm at all times except
in the presence of the King. 2. (sl.) inferior; of poor quality:
I read no matter how bad the book and some are right under
the arm.
under-arm —(attrib.) 
1
. that can be carried “under the arm”:
Whatever she carries about with her she keeps in an “under-
arm bag.” 2. (sl.) pornographic; obscene: One of the first
things Randle asked him was did he make any under-arm
films.
under the color of something • under the colors of
something
under the color of something —under pretext or pretence of
smth.: A lot of what this unscrupulous politician writes under
the color of truth is nothing but a pack of lies.
under the colors of something —supporting a particular cause,
movement, or country: Northmour was killed fighting under
the colors of Garibaldi for the liberation of the Tyrol.