HI Tim
I had double checked this. The user name of the user, who is logging on to Win 7 PC is exactly the same as user in SAP (su01). Exactly the same is in DE.NNNNNNNN.NET AD. No difference at all.
In table USRACL i have:
PNAME = p:mxxxxx@DE.NNNNNN.NET - exactly the same way as in us01.
One more update. I read in your answer to other poster, that there should be transitive communication between domains - we have NOT configured that. Our AD-admin don't think this is needed, when we have two-ways communication between domains.