"We decide to engage a lawyer for the case."
"We decide to hire a lawyer for the case."
Is engage used particularly in British English? Do speakers of American English use engage in such examples?
Simply put, engage is really not all that common when it comes to hiring people. Just use hire; its primary definition is more applicable to the situation than engage. It is interesting to note that the definition of hire includes the word engage. I think that hiring someone is a more specific form of engagement, and if you're looking to achieve clarity, then hire is your best bet.
I'm fairly certain that the definitions of these two words have little variation between British English and American English, so this answer most likely applies to both.
Edit: Colin Fine has informed me that this does not necessarily apply to British English.
The meaning of the words given by the NOAD are the following:
Similar definitions are given by the Merriam and Webster, which gives the following definitions:
Looking at the CoCA and BNC for sentences containing "engage a lawyer" and "hire a lawyer" I get the following results: