Droplet running WP and Exim won't email local FQDN
I am running a Droplet with Ubuntu 14.04, WordPress and Exim for email.
I have setup Exim to be able to receive email notifications from WordPress and everything appears to be working fine as long as I use an email address that is NOT on the FQDN.
In other words, if my site is example.com, I can receive emails with no problem as long as those email are sent to firstname.lastname@example.org or email@example.com but, if I try to send to firstname.lastname@example.org, the message doesn't get delivered.
This issue not only happens from within WordPress, but also from the command line.
If I enter
echo "This is a test." | mail -s Testing email@example.com the email does not get delivered, however if I try
echo "This is a test." | mail -s Testing firstname.lastname@example.org the message gets delivered without a glitch.
I can send email from any email client to email@example.com without any problem so, evidently, it is something related to the configuration on my droplet.
Any help with this will be much appreciated.