SAN FRANCISCO — Gov. Jerry Brown of California on Monday ordered a limited retesting of evidence in a highly contested murder conviction.

Kevin Cooper was convicted in 1985 of murdering Douglas and Peggy Ryen, their daughter and a neighbor’s son in Chino Hills, an affluent suburb of Los Angeles. His lawyers had petitioned the governor for clemency in 2016.

Mr. Brown said in July that he would consider additional testing in the case, and on Monday he ordered the retesting of a shirt, towel, hatchet handle and hatchet sheath.

In his order the governor said that the purpose of the testing, which will be carried out under the supervision of Daniel Pratt, a retired judge, was to see whether DNA from another person in the F.B.I.’s database is present on the items.