How do you know for sure whether or not to use a closing? One good test is to ask yourself whether this email is more akin to an instant message/text or a business letter. If you're giving your teammate a quick update on an ongoing project, a formal closing might not be necessary; if you're throwing your hat in the ring for a promotion, it definitely is required. When all else fails, and you're still not sure, err on the side of caution and include it. You'll never go wrong by being too polite and respectful.
Similar to business or employment_related correspondence, military correspondence also has set standards for the closing (also called a “valediction”) which one should present before one’s signature. If your company conducts business with the military – or if you are applying for a job in the military – you should be aware that “Very Respectfully” (often abbreviated as “V/R”) is used in written and emailed correspondence between military members.