Hi experts!

My SF Embedded Login that uses Salesforce's Community as Identity Provider is working fine in my website.  However, if a user bypass my website and log in into the community directly via SF's standard login screen, then navigate to my website within the same browser, then the Embedded Login doesn't know how to detect the logged in user in this scenario.

How can the Embedded Login detects logged in user?

Note that I've found that the session id is stored in sid in the browser's cookies, but I can't access it because my website and SF are on different domains.