To challenge the Johto Elite 4 again, you need to talk to the NPC Silver below the market in the Pokemon League, he will challenge you to a fight.
After the successful fight, he will tell you that he is waiting for you at Dragons Den to challenge the greatest dragon masters the Johto region has.
You must now make your way to the Dragons Den in Blackthorn City (don’t forget HM Surf and Whirlpool).
Inside the Dragons Den you have to enter the Masters house and leave above it, there Silver will be waiting for you, Claire and Lance will also join you.
You will now fight with Silver in a team, only the first 3 Pokemon of your team will enter the fight.
If you are successful in the fight, Lance will give you the key for the rematch of the Johto Elite 4.
Now fly back to Pokemon League and find the NPC Liu Yu Te on the upper floor, he will give you access to the Johto Elite 4.
Just go down the stairs again and then talk to the guard of Johto Elite 4 on the left side who will give you access.
Here are the teams of the Elite 4:
BRUNO
KOGA
WILL
KAREN
LANCE