Can someone write "we embraced" to mean "we embraced each other?"
Yes, it's very common to use that form of expression to imply that you do something with/to each other. For example, we met, we talked, we shook hands, we played golf, we hugged, we kissed.
It is perfectly well to write "we embraced" to mean "we embraced each other".
The simplest sentences in English contain a single independent clause. That means that they:
- Contain a subject
- Contain a verb
- Express a complete thought
Apply those tests and you'll see that the answer is "yes": your sentence has a subject ("we"), a verb ("to embrace" in the simple past tense) and conveys a complete thought (i.e. is not dependent on any other clause to complete the meaning).