We can go back and forth for days here, even when you spell out the exact commands!
Its easier for someone to just teamviewer in and fix it for you in like 10 seconds.
As already stated YOU NEED to be an OWNER to set permissions.. Then set them as you need - yes by default admin and system should have full control.
What you do after that on a non system drive is up to you. I root of my D set like this
Keep in mind when your in explorer your not normally elevated - so your not ADMIN, does not matter if your in the admin group or not.. This is for example you get a prompt to write a file to root of C
Notice the permissions on your C drive -- users and authenticated users have what?
If you don't want that to happen then you need to set the permissions how you want them, to set permissions you have to have full control already or be an owner, or be able to elevate to admin that had permissions, etc. etc..
Now if you want we can have yet another discussion on the basics of NTFS permissions.. But the information to fix your issue has already been given.
As stated take OWNER of your D from the root, write this ownership down the tree. Then set the permissions how you want.. Since the disk came from other OS, then sure its a good idea to reset them! Its not going to hurt anything if its just a data drive. You could set everyone to full control as the only permission if you wanted.. Its your DISK, set the permissions how you want!
Or just PM details so I can TeamViewer in and we can straighten all out in like 10 seconds.