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ITV News has obtained documents that suggest the UK Security Services were trying to recruit Michael Adebolajo and had also approached members of his family and urged them to get him to agree.
ITV News UK Editor Lucy Manning reports:
The family of Michael Adebolajo, one of the men arrested over the murder of Drummer Lee Rigby, today expressed their "profound shame and distress" over the "senseless killing" and sent their "heartfelt condolence" to the soldier's relatives.
One of the men shot by police in the wake of the murder of soldier Lee Rigby has been discharged from hospital, Scotland Yard said.
The 22-year-old, understood to be Michael Adebowale, from Greenwich, south east London, was taken into custody at a police station in south London this afternoon.
He was arrested on suspicion of the murder of Drummer Rigby on May 22, and this afternoon was further arrested on suspicion of the attempted murder of a police officer.
He will now be interviewed by detectives from the Metropolitan Police Service Counter Terrorism Command.
The name of murdered soldier Lee Rigby will be inscribed on a national memorial honouring Britain's fallen servicemen and women.
Trustees for the Armed Forces Memorial at the National Memorial Arboretum, Staffordshire, said Drummer Rigby's name would be included as he had died in a terrorist attack.
Drummer Rigby, 25, who had been serving with the 2nd Battalion the Royal Regiment of Fusiliers, was killed in a knife attack carried out in broad daylight in Woolwich, London, last Wednesday.
In a statement, the memorial trustees said:
"Lee Rigby was killed as a result of terrorist action outside Woolwich Barracks in London on May 22 and qualifies for inclusion on the Armed Forces Memorial.
His name, along with other members of the UK armed forces who are killed on duty or through terrorist action in 2013, will be engraved in time for the annual service of dedication in mid-2014.
The families of the bereaved are always invited to attend, usually in the presence of a member of the Royal Family, a defence minister and senior armed forces representatives."
The memorial honours all members of the UK's armed forces who have died in conflicts or as a result of terrorist attacks since the end of the Second World War.
Following calls to tackle hate speech at universities, Business Secretary Vince Cable said there was a "very difficult balance to strike" between protecting free speech and stopping extremists inciting violence.
Speaking at an event in central London, Mr Cable said: "We have recognised that there are extremists on university campuses. We have got a strategy which we are developing with the university authorities to deal with it.
But it's obviously a very difficult balance to strike between freedom of speech, which is absolutely fundamental to our value system as a country, and stopping incitement to violence.
Getting that balance right is very tricky."
MI5 is not in the dock even though MPs are preparing to investigate the service's operations prior to soldier Lee Rigby's murder, a former defence secretary said today.
Parliament's Intelligence and Security Committee (ISC) will also answer whether or not security agencies fell short in the case of the Woolwich murder, its chairman Sir Malcolm Rifkind said.
Sir Malcolm added that the effect of Government spending cuts on the security services will be analysed during the ISC's review and any further reduction in resources would be a cause for concern.
Speaking on Radio 4's Today programme Sir Malcolm said,
"Do remember one fundamental point: the fact we have not had anyone killed until these tragic events in Woolwich since the 7/7 bombings (in London) in 2005 is not because there hasn't been terrorist plots.
Every year since 2005 there has been at least one, sometimes two or even more, terrorist plots which were disrupted and prevented from killing British citizens, partly because of the work of MI5 - in some cases very largely because of the work of MI5 - and other intelligence agencies."
Police are still questioning a 50-year-old man on suspicion of conspiracy to murder. He was arrested yesterday in Welling in South east London, following the killing of Drummer Lee Rigby in Woolwich last week.
Latest ITV News reports
Documents suggest Security Services were trying to recruit Woolwich suspect Michael Adebolajo and had also approached members of his family.
Police will today question one of the men suspected of murdering Drummer Lee Rigby in Woolwich, a week after the soldier's death.