The car is new to me and I’d naively thought, since they’re all the same, that they were just an unusual fitting. I ordered a spline socket set but none of those fit.
I’m fearing that they might be locking wheel studs (is there such a thing?) and that I don’t have the correct fitting to get them off.
Raising the ride height is proving more challenging than I’d expected!
Possibly Tuner wheel nuts?
Yes, they’re tuner type wheel bolts. They look like these ones to me, which have a 10 point star key:
So the seller should have really included this with the sale of the car! Supposedly the star key gives some added security, but in reality they’re just more annoying to get off. Most people fitting them do so for appearance purposes (…in my opinion!!).
no most people fit them on aftermarket wheels because the socket doesn’t go into the wheel without scratching it!
Hmmm. A properly protected socket achieves the same, without an ungainly protrusion… but fair enough!
Something like these?
Thank-you. At least I know what I’m working with now. The guy I bought a it from was a nice chap - I’ve messaged him - fingers crossed. I’m sure it was just on oversight.
Ride height is a bit of a nightmare - it’s incompatible with National Trust driveways 
Yes, I was referring to the photos uploaded by the OP.
