Virtualbox: only 32bit guests possible even though virtualization enabled in BIOS / Intel Process Identification Utility shows opposite to BIOS virtualization setting

Virtualbox on my Windows 8.1 stopped running 64bit guests a while ago. I did not track down this problem. Now some months later i tried again and found some confusing things.

First setting:

BIOS virtualization enabled
Intel Processor Identification Utlility in 8.1: virtualization disabled

Second setting

BIOS virtualization disabled
Intel Processor Identification Utlility in 8.1: virtualization enabled

With both settings: Virtualbox runs 32bit guests but no 64bit guests.
 

After some searching, i realized, what was happening:
I added Microsofts Hyper-V virtualization. With that enabled Windows 8.1 is no longer a real host. It is just another guest (the most important guest) on this computer. So with Hyper-V enabled i was trying to run Virtualbox inside an already virtualized Windows 8.1.
After that it was easy: Just disable Hyper-V on Windows 8.1:

And after a restart of Windows 8.1 i was able to run 64bit guests on Virtualbox again....