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Islamic State has claimed it was behind the Westminster attack in a statement issued by its Amaq news agency.
In a tweet, the agency said: "A soldier for the Islamic State carried out the operation in answer to calls to target the people of coalition states."
The phrasing suggests the attacker was inspired by Islamic State, and had not been directly trained by it.
Theresa May has told Parliament that the perpetrator was a British citizen who was known to police and security services.
In a statement in the House of Commons, the Prime Minister said the attacker was a "peripheral" figure who had once been investigated by MI5 over concerns about violent extremism.
"His identity is known to the police and MI5 and when operational considerations allow, he will be publicly identified," she said.
"What I can confirm is that the man was British-born and that some years ago he was once investigated by MI5 in relation to concerns about violent extremism.
"He was a peripheral figure. The case is historic. He was not part of the current intelligence picture.
"There was no prior intelligence of his intent or of the plot."
Meanwhile, Sky's senior political correspondent Beth Rigby tweeted: "Minister tells me assailant got into New Palace Yard cos gate was open: person leaving was Acting Police Com Craig Mackey."
Eight people have been arrested during raids in London, Birmingham, Brighton and Wales that were linked to the terror attack.
Four people were killed in the attack before the assailant was shot dead.
The victims are: PC Keith Palmer, a member of the Parliamentary and Diplomatic Protection Squad; a 43-year-old mother Aysha Frade; American Kurt Cochran, who was visiting London; and a 75-year-old man who had his life support switched off in hospital on Thursday evening.
A family statement said that Mr Cochran and his wife Melissa "were in Europe to celebrate their 25th wedding anniversary. Melissa also received serious injuries in the attack, and is being cared for in the hospital".
Acting Deputy Commissioner Mark Rowley, the Met Police's senior anti-terror officer, said 29 people had been treated in hospital, with seven people in a critical condition.
He added that the attacker, named as Khalid Masood, is believed to have been working alone and was inspired by "international terrorism".
Mr Rowley said there was no specific information to suggest any further threat to the public.
"We must not allow terrorists to create discord, distrust and fear," he said.
Three police officers who were injured by the vehicle are in a stable condition in hospital, according to Met Police Federation chairman Ken Marsh.
Armed police raided a flat in Hagley Road, in the Edgbaston area of Birmingham, on Wednesday night.
A witness who works nearby told the Press Association: "The man from London lived here. They came and arrested three men."
Dozens of officers, some armed, were pictured outside the second-floor flat, above a parade of shops, which was cordoned off by police.
Roads around the home were initially closed off, but later reopened although officers remained at the scene.
Car hire firm Enterprise confirmed the Hyundai car used by Masood was one of its vehicles, reportedly rented from its Solihull branch.
It said an employee had identified the vehicle after seeing the number plate online.
Masood, who was armed with two knives, drove into pedestrians on Westminster Bridge before crashing into railings in front of Parliament at around 2.30pm on Wednesday.
He was shot dead after going through a gate towards the Palace of Westminster and attacking PC Palmer.
Following the attack, the Prime Minister said the UK's terror threat level would remain at severe.
Mr Rowley said leave had been cancelled to allow for an increased police presence in London.
The House of Commons and House of Lords sat at their normal times on Thursday.
MPs in the Commons stood to observe a minute's silence at 9.33am - a time chosen to commemorate PC Palmer, whose shoulder number was 933.
Defence Secretary Michael Fallon said the working assumption is that the Westminster attack is linked to Islamic terrorism.
He said security arrangements at Westminster will be reviewed to look at whether they are "adequate" and whether police at the front gates should be armed.
In a message to the Metropolitan Police's Acting Commissioner, Craig Mackey, the Queen said her "thoughts, prayers, and deepest sympathy are with all those who have been affected by yesterday's awful violence".
Sadiq Khan, Mayor of London, told Sky News: "It's not the first time terrorists have attacked our city; it's not the first time, I'm afraid, Londoners have tragically lost their lives and been seriously injured; it's not the first time people have tried to divide our communities and destroy our way of life.
"But I tell you what, in the past we have risen, we've shown our resilience, we've returned to work and returned to normality. We have done it in the past and we will do it again now."
Andrew Parker, director-general of MI5, said the security service's operational response had been "fully mobilised".
He said: "The thoughts of the men and women of MI5 are with the families of those killed in Westminster yesterday, and with the other innocent people injured in this appalling and disgusting attack."
Earlier reports had said five people, including the attacker, had been killed, but police confirmed a total of four on Thursday morning.