Hello Sebastian,
If you can get into Windows 7 there are some tools that can be used to retrieve a product key. ( Third party tools, not Microsoft tools).
Newer releases of Windows store the product key online tied to your MSA account.
But as far as I know there isn't a way to retrieve a product key using the serial number of the machine nor can it be regenerated from teh first 10 digits of the product key.
Sorry
Darrell