sending and receiving emails with domain aliases on GSuite

One of our units had two domains, longname.tld and shortname.tld - they asked if they would receive emails sent to user@shortname.tld which had been set up as a domain alias in GSuite. My reply - 

The above holds good for receiving emails from username@shortname.tld

If you wish to send emails from username@shortname.tld (as against username@longname.tld), you need to follow the procedure below by logging on to username@longname.tld 

(from https://support.google.com/mail/answer/22370?hl=en)

1. On your computer, open Gmail logged in as username@longname.tld .
2. In the top right, click Settings   Settings.
3. Click the Accounts and import or Accounts tab.
4. In the "Send mail as" section, click Add another email address.
5. There, enter username@sssvv.org
   

You also have an option to make whichever email you wish to have as the default sending email, just below that section.