Use a good tight bronze brush, soak it in BBS and give the bore 10 strokes, resoak, give it 10 more strokes, resoak and give it ten more. Then run a couple tight patches through the bore and check for copper residue on the muzzle with a good bore light. More then likely there will be none there. IF there is repeat the process and recheck.
When you run the second dry patch through the bore it should feel smooth as glass going down the bore. Pay attentoin to how the patch feels just ahead of the throat as this is where copper will generally collect if it is going to. Should not be an issue with the Lilja barrel.
One last tip, make sure you oil your bore before shooting the next two rounds. Any bore that is completely dry will copper foul on the first shot. I highly recommend getting a high quality cutting oil and coat the bore good adn then run a semi tight dry patch through to remove the excess oil. There will be a light film on the bore when you fire again but this seems to greatly aid in smoothing up the bore quicker which again is not an issue with the Lilja barrel.
You will notice that around shots 6 to 8 the barrel will simply stop fouling in most cases. When you see this do another couple two shot pairs and then you can skip to the three shot strings to speed things up.
BJs barrel really never did foul so he could probably go straight to the 3shot strings if he wanted.
The oil though is the best thing to use to aid break in.
Also keep those bore brushes tight, you will go through several of them in the break in process but if they get loose in the bore they will not clean as quickly so keep them snug and replace when needed.
Remember to clean that chamber, bolt locking lug area and muzzle brake
after each scrubbing. A brush bristle blown out of a muzzle brake can be a serious problem if it lands in the wrong spot. Clean them out totally before shooting.