I have located the problem precisely but I don't know how to solve it.
When I'm on my custom cms page with my new admin login form, Magento gives me a different key_form than my default admin login page.
So when I try to login with my custom form, Magento redirects me to my default admin login page and warns me with the following message
Invalid Form Key. Please refresh the page.
My Frontend Custom CMS Admin Login Page:
<input name="form_key" value="cbzRGF9ccU8Ac8Ib"/>
My Backend Default Admin Login Page:
<input name="form_key" value="B9O7srzwq7utjQ79"/>
How to generate a form_key on my custom Login page that will be the same as my default admin login form (www.website.com/admin).