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If using a classic profile, you can go to the 'profile options' page and the box for the YouTube URL is right under the theme override box on the right-hand side of the screen. You just use the web URL to the YT page with the desired video and Gaia will take care of the rest.
If you want it to autoplay, add &autoplay=1 immediately after the code.
If you're using a custom profile, make sure it has a place for multimedia. Otherwise, posting the code will do nothing.
If using a new profile, I BELIEVE you take the URL, go to Profile > Edit, click Content > Media, then paste the URL in the bar that pops up.
I don't use the new profiles at all so I'm not entirely sure about the steps to post with V2.
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