Header, head and boddy tags using the HTML editor

I use the HTML editor to create HTML text, but the editor adds the following at the beginning:
<!DOCTYPE html>
<html>
<head>
</head>
<body>

and at the end:
</body>
</html>

But I want to create paragraphs instead of full pages, so that I can concat all my HTML blocks using a php class to a complete page. I only want to save the text between the body-tags.
Is there a way to make the editor not creating thes page and body tags?

It seems to happen only when using an editor template, when the standard editor is used you don’t get these tags (but then you have to configure the toolbar in every form you use the editor).