Why does a browser in a different domain not respond at all to “WWW Authenticate : Negotiate” header sent by mod_auth_kerb?

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Apache

SSO

I have implemented SSO through mod_auth_kerb in our apache-active directory environment and it works just as expected. However the following knowledge is bugging me :

I requested a Kerberos protected page from two client machines, one user belonged to the Kerberos-setup domain and the other user belonged to some other domain. I then compared the HTTP packets on the two machines. On both the machines, after the request for the Kerberos protected page is sent, the server responds with the following HTTP packet :

HTTP/1.1 401 Authorization Required Date: Wed, 05 Sep 2012 14:25:20 GMT Server: Apache WWW-Authenticate: Negotiate WWW-Authenticate: Basic realm="Kerberos Login" Content-Length: 60 Connection: close Content-Type: text/html; charset=iso-8859-1

However, after the above response from the server the client machine's browser belonging to the Kerberos-setup domain responds with a WWW-Authenticate : Negotiate 'token', whereas the other client browser(user belonging to some other domain) does not respond at all.

Now my understanding is, that the client belonging to the other domain should have also responded with its own TGT+Session key token, which the active directory should have rejected. But why this client does not respond at all to the server's WWW-Authenticate : Negotiate challenge is beyond my logic. What is even more confusing is that the server's HTTP response(given above), does not contain any information about the domain it is linked to.

So on what basis is the client browser belonging to the correct domain decide that it has to respond to the server's WWW-Authenticate : Negotiate challenge, and on what basis does the client belonging to some other domain decide not to respond to the same ?

Note : Both the client machines have Windows 7 and active directory is a Windows 2008 server.

I am trying to understand mod_auth_kerb's implementation of SSO, and this particular knowledge is key to that.