The trial "confirmed that the deceased US service members followed all established procedures when accessing the base the day of the incident, as we have noted before", the US Embassy in Jordan said.
Last month, however, Abu Tayeh was finally charged and convicted of premeditated murder in the killing of all three, although he pleaded not guilty at a military court in Amman.
According to Jordanian news outlet al-Ghad, the soldier who shot three U.S. military trainers last November in al-Jafer airbase in Ma'aan governorate has been sentenced to life imprisonment by a Jordanian military court.
The US soldiers' family members were in court to hear the verdict on Monday.More news: Collins points to an underappreciated flaw in the GOP health plan
In addition to extradition, the families are also asking for "military or criminal charges against all the Jordanian military gate guards and the barrel truck operators for dereliction of duty for failing to protect the lives of our soldiers or cowardly conduct under fire".
Tuwayha - who is not thought to have links to any militant groups - told the military court in the capital Amman he had opened fire after hearing gun shots.
Despite the conviction some relatives criticized Jordan's handling of the case and are disappointed al-Tuwayha didn't receive the death penalty, saying the maximum sentence in this case, life in prison, was not enough.
The US Embassy said, "We are reassured to see the perpetrator brought to justice".More news: Consultant slams doctor's 'magic potion' treatment for Charlie Gard
"Despite this tragedy, Jordan remains a strategic partner", Barbee added.
The three US Army Green Berets were killed on November 4, as their convoy came under fire while they were entering an air base south of the country.
Jordan initially tried to shift the blame to the United States, saying the Americans triggered the shooting by disobeying entry rules, a claim that was later withdrawn. But US officials disputed this and a subsequent US military investigation found surveillance footage of the event.
Al-Tuwayha has said he had "no intention of killing anyone" and felt no resentment towards Americans.More news: People Ejected From Out-Of-Control Boat In Indiana, Operator Arrested
In 2015 a Jordan police captain opened fire at an officer training center killing two Americans and several others. The third man, Staff Sgt. James Moriarty, suffered a mortal wound after firing his pistol at the Jordanian guard.
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