The allegations stem from explosives found at the family's home, where the mother ran a licensed daycare facility, said sheriff's Sgt. Dennis Gutierrez.
Benjamin Kuzelka, 23, walked into a hospital and told emergency personnel that he had shot his own hand, said sheriff's Sgt. Gutierrez.
An examination of the wound determined it was not consistent with a gunshot, and Kuzelka later told deputies that he had been injured by explosives, Gutierrez said.
Sheriff's deputies, along with agents from the FBI and Bureau of Alcohol, Tobacco, Firearms and Explosives, went to his home at 30524 Audelo Street and found bomb-making materials and marijuana plants, the sergeant said.
Kuzelka was in serious condition and will be arrested when he recuperates.
His brother, Grey Timothy Kuzelka, 21, was arrested and booked into the county jail. He had been free on bail since Sunday, following an arrest on suspicion of drug possession, sales and transport.
Their mother, Rebecca Kuzelka, 55, was also being held.
The possibility that the family is associated with hate groups or other radical home grown or international terrorist with intentions of bombing southern California targets is being investigated by local, county and federal authorities according to an investigator.
Law enforcement in surrounding counties including Orange and L.A.Counties are investigating to see if there is any connection between the Kuzelka family and any recent bomb threats.
LAX has long been believed by terrorist experts to be a high grade target for terrorist.
In a wide-ranging Rand report it considered the potential casualties from car bombs, mortars, snipers and surface-to-air missile attacks the most likely scenario and is the first public blueprint of the airport's greatest vulnerabilities.
LAX, the world's fifth-busiest airport, is considered the state's top terrorist target. An Al Qaeda plot to explode several luggage bombs in terminals was foiled in December 1999.
According to former president George Bush a Los Angeles skyscraper was a terrorist target.