Meaning: have a private or selfish reason for doing something. Have an ulterior motive.

Example Sentences:
1.- He should not become the chairman of the committee as he has too many axes of his own to grind.
2.- When I see him strongly supporting someone who could be his rival, I cannot help but think that he has an axe to grind.

Meaning: with the backing, support, patronage of.

Example: Financial aid is being provided to the country under the auspices of the World Bank.

Meaning: water. A joking way to refer to a glass of water.

Example Sentences:
1.- It is blistering hot outside, I could really do with an iced cold glass of Adam’s ale.
2.- Unfortunately, we have no wine or beer in the house. You will have to settle for some Adam’s ale.

Meaning: keep distance from something or somebody. Avoid intimacy or familiarity

Example Sentences:
1.- I like to keep my office colleagues at arm’s length outside of office. I don’t want them in my personal life.
2.- He always had the feeling that she was keeping him at arm’s length.