The government offered drillers lucrative “royalty relief,” a holiday from the standard 12 percent royalty, as an incentive to increase exploration and production in waters that were thousands of feet deep.

But as a result of what investigators believe was bureaucratic bumbling, the leases for 1998 and 1999 omitted a standard escape clause that rescinded the special inducements if oil prices climbed above $34 a barrel.

Midlevel officials at Interior spotted the omission in 2000, and quietly made sure to include the escape clause in all subsequent leases. But no one tried to fix the leases that had already been signed, and almost no one talked about them until oil prices started to climb above $34 a barrel in 2004.

At a hearing of the House Government Reform Committee last September, Ms. Burton testified that she had first learned about the problem in late 2005 or January 2006. Ms. Burton and other senior officials publicly confirmed the error in February 2006, after The New York Times published an article about it.

But investigators have unearthed a series of e-mail messages by officials working under Ms. Burton in March 2004. At the time, energy prices had recently climbed above the thresholds that were supposed to stop the incentives, and oil companies were pressing the Interior Department to confirm how it would treat the leases from 1998 and 1999.

Marshall Rose, chief economist for the Minerals Management Service, wrote the agency’s associate director at the time, Thomas Readinger, that the decision had to be made by the “directorate” — Ms. Burton and her top deputies.

Mr. Rose told Mr. Readinger that he believed the leases entitled companies to the incentive regardless of oil price levels, and that he had told his own subordinates that “you and the director were aware of the need to make a decision on this matter.” Mr. Readinger, who retired last year, responded to Mr. Rose a few hours later by writing, “Sounds like we have an answer. Let’s go with it.”