Home Secretary Theresa May has said the murder of soldier Lee Rigby "united the entire nation in condemnation".
Michael Adebolajo, 29, and Michael Adebowale, 22, were found guilty of the murder of Fusilier Lee Rigby near Woolwich Barracks in May.
"The sickening and barbaric murder of Drummer Lee Rigby united the entire nation in condemnation and I welcome the jury's decision," May said.
"But we must not forget that this appalling and public act of violence and terror also robbed his family and loved ones of a brave, young man with his life ahead of him. My thoughts are with them at this difficult time."
In 2010, Michael Adebolajo visited Kenya and is believed to have been preparing to train and fight with al-Qaeda-linked militants in Somalia.
According to intelligence files seen by ITV News, he travelled to the country's coast and met a group under the surveillance of anti-terrorism police.
It is claimed they contacted a radical cleric for help to get to Somalia. That cleric told ITV News correspondent Rohit Kachroo: "Michael was a very nice young man - a man of strong character and if you look at him you believed in his cause."
"They were to join their fellow Mujahideens [Muslims who proclaim themselves warriors for their faith] in Somalia and to assist, put a stop to the oppression many of the Muslims in Somalia are going through."
Richard Taylor OBE, father of murdered schoolboy Damilola Taylor, mentored one of Fusilier Lee Rigby's killers as a young boy. He believes Michael Adebowale was radicalised during a period in jail.
He told ITV News: "I thought his going to prison would have reformed him and he'd come back and get back into the community with positive behaviour."
"Something must have gone wrong in prison, they must have indoctrinated him in the wrong way", he added.
Lee Rigby was the victim of one of the most savage offences ever prosecuted, the Crown Prosecution Service said.
Sue Hemming, Head of Special Crime and Counter Terrorism at the CPS, said: “The murder of Fusilier Rigby was brutal and its perpetrators carried out one of the most savage offences ever prosecuted by our counter terrorism lawyers.
“As a soldier, this young father had dedicated his life to keeping people safe, including from the threat of terrorism. That dedication to his country cost him his life and was in stark contrast to the appalling conduct and extremist views of the men who murdered him.
“We recognise that this trial has been exceptionally difficult for Lee Rigby’s family but I hope they can take some limited comfort from the justice achieved today and the fact that both defendants now face a very long stay in prison.”
The death of Lee Rigby "brought people together", Assistant Commissioner Cressida Dick, the head of the Met’s Specialist Operations said today, after two men were found guilty of his murder.
This horrific attack, which took place in broad daylight on the streets of London, shocked the country and was intended to divide communities. It had largely the opposite effect and has, in fact, brought people together.
The court heard of extraordinary acts of courage and compassion from members of the public at the time of this terrifying and dreadful incident.
The family of Lee Rigby have said "no amount of justice will bring Lee back".
Members of the Rigby family cried outside the Old Bailey as a statement was read by Detective Inspector Pete Sparks, following the convictions of Michael Adebolajo and Michael Adebowale.
The wife of Fusilier Lee Rigby killed in an attack in Woolwich in May has said his memory "lives on". Speaking outside of court after two men were found guilty of the murder, Rebecca Rigby said:
This has been the toughest time of our lives and no-one should have to go through what we have been through as a family.
These people have taken away my baby’s Dad but Lee’s memory lives on through our son and we will never forget him.
I now want to build a future for Jack and make him proud of his Dad like we all are.
Trained first aider Vikki Cave went to help Fusilier Lee Rigby when she saw him lying on the Woolwich street and called the police.
Speaking to ITV News UK Editor Lucy Manning, she said: "It was a natural reaction to go and help and I didn't realise that they [Lee Rigby's killers] were there and when I did realise, that was a bit of 'Oh my god, it’s kind of scary’, especially when I saw the gun."
Ms Cave said a woman who was sitting beside the soldier rubbing his back was "so, so brave, absolutely amazing. If that was my son I'd want someone to be there. She was incredible."
"I spoke to the man and said: 'Is she ok, are we safe, are you going to hurt us?' And he said the women and children are safe but when your police and soldiers get here you need to back off as I can't guarantee anything."
Lee Rigby's family said they were satisfied that justice had been done and said: "This has been the toughest time of our lives. No one should have to go through what we have been through as a family."