Asp.net Identity (MVC 5) cross subdomain id and application pool id

I used the method provided in ASP.NET Identity Cookie via subdomains to implement cross-subdomain registration.

For example, I have two sites: account.mydomain.com and www.mydomain.com. Users who are logged into account.mydomain.com are also logged into www.mydomain.com.

However, there is a strange problem: I have to change the "Application Pool ID" to "NetworkService" or "LocalSystem" other than the default "ApplicationPoolIdentity". Or, even if someone is logged into account.mydomain.com, they cannot log on to www.mydomain.com.

I wonder what the reason is, and is this the issue that changes the identity on the NetworkService?

Thank you so much!