The long-delayed bill to put teeth in the Fair Housing Law took a large step forward yesterday as a result of the Republicans' new interest in the activities of Billy Carter.

Most Republicans on the Senate Judiciary Committee have shown little interest in speeding the bill, which would provide machinery to enforce the 1968 law forbidding racial discrimination in housing. They blocked subcomittee action for months by not showing up to make a quorum. But lately they have developed an extraordinary desire to see what they can turn up about the activities of the president's brother as an agent for the Libyan government.

Yesterday's committee agenda included the fair housing bill and a request from Sen. Bob Dole (R-Kan.) to immediately rev up a Carter investigation. fDole tried to take up the Carter matter right away, but Democrats wouldn't let him open his mouth on that subject until Republicans agreed that the committee would meet, mark up and report out the housing bill next week. If not enough members show up to make a quorum they can be polled on their positions on amendments.

Once the agreement to vote on the housing bill was nailed down, they set up a four-member subcommittee to decide how best to proceed on the Carter matter.

Civil rights groups believe they have the votes in the full Jidiciary Committee to reverse actions by the subcommittee and set up strong and speedy enforcement machinery in an administrative office rather than in the federal courts.