The only way you can be sure of a 'pure' Labrador is if you buy from a breeder who registers pups with the kennel club and if you trust that breeder. If not then you will never really be certain of anything, that includes health testing not just pedigree. If it doesn't bother you whether she's 'pure' Labrador or not, then just enjoy her for the character she is. There are no ways of telling whether she's 'pure' or not, and if she is, and the breeder didn't bother to register pups then there are lots of reasons they either didn't bother or even couldn't. Some years ago now I used to be pretty much stalked by someone with a 'pure' bred Labrador dog, he wanted to eventually stand at stud, but he wasn't registered with the KC because the breeder couldn't register the litter. Back then you could register up to 8 litters from a bitch and this bitch had had more litters than that even, hence that pup not being registered but being sold as a pure bred Labrador. Ignore the myth about the wave down the back indicating they are a well bred Labrador, it's nothing more than an old wives tail. Lots of dogs have wavy fur down their back because it's just a bit longer, that's the only reason it's wavy, nothing to do with how the dog is bred.