LDS Instances like Active Directory also have a Configuration Partition and a Schema Partition. On TechNet it is documented:
By default, the security principal that you specify as the AD LDS administrator during AD LDS setup becomes a member of the Administrators group in the configuration partition.
How do I get to the configuration partition in an LDS Instance?
Well if I connect to my application partition created during the LDS installation wizard and go to my Administrators, I can see nested inside is another Administrators group residing inside the Configuration Partition.
I can now connect to this using ADSIEdit.
Under the administrators group in the configuration partition I can find the account used when I installed the LDS Instance as per Microsoft documentation on TechNet.