I made 2 tests with bitaddress.org (livenet)

WIF = 5JoVxXdyoVFPaVL4xHQUvdb8vVRDFCdeh8A16fodXMiYKmrgtCu
        = 1KLgFamwn7cm1qQSnuFVuYw99dmWUVZC9b

WIF = 5K1xnQXX5d1xVaRab75HbLT6oPejddN1eUdesL8epMdjRgHVuqG
       = 1Npcxoh99nGGf829uJhvMWan4vUbTzfi9y

If I understand well, the private key is the same in livenet and tesnet:
the private key is just a ramdom number and the WIF format will
also be the same.

Then we calculate the by adding points on a elliptic curve,
so the pubkey is a pair of coordinates (x,y). Then we write the pubkey
like "04 x y" OR "02 x" (if y is positive) OR "03 x" (if y is negative).
It is only when we create the base-58 pubkey that we use a different
format so the pubkeys looks different in testnet and livenet.

Is it correct ?