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Default WordPress WP Editor removing style tags and html tag

I am working inside the WP Admin area and there is a textarea that is using the default WP Editor, not TinyMCE. The WP Editor is removing the <style> tag and also other HTML tags. I am calling it WP Editor, but maybe it’s official know as something different? It’s the editor with the two tabs in the right corner (Visual,Text)

Is there a way to prevent this Editor from removing all HTML tags?

The content I am adding in the WP Editor is being used in outgoing emails so it’s really a full HTML document which is why I am trying to preserve the formatting and html tags.

I notice when I switch from the "Text" tab to the "Visual" tab many HTML tags are stripped.