A federal jury will hear opening arguments Sept. 4 in a high-stakes lawsuit that accuses State Farm of secretly bankrolling the campaign of an Illinois Supreme Court Justice who then cast the deciding vote to overturn a billion-dollar judgment against the insurer.

The lawsuit pits State Farm Mutual Automobile Insurance Company against a group of policyholders who say the insurance giant broke racketeering laws by secretly funding the winner of the most expensive state supreme court campaign in U.S. history—and then lying about that backing so the justice wouldn’t have to recuse himself from the appeal of a $1.05 billion...