The third London Bridge attacker has been named as 22-year-old Youssef Zaghba, a Moroccan-Italian man.
Pakistan-born Khuram Butt, 27, and Rachid Redouane, 30, both from Barking, were the other two attackers.
The trio drove a van from London Bridge into pedestrians then got out and began stabbing people in an attack that left seven people dead and 48 injured.
Armed officers killed all three within eight minutes of receiving a 999 call.
Two more of the people killed in the attack on Saturday night have also been named - Australian nurse Kirsty Boden, 28, and French national Alexandre Pigeard, 27.
Metropolitan Police said a 27-year-old man had also been arrested in Barking in connection with the investigation on Tuesday. The 12 people arrested shortly after the attack had already been released without charge.
The Islamic State group has said its "fighters" carried out the attack.
The Metropolitan Police said Butt had been subject to an investigation in 2015, but there had been no suggestion this attack was being planned.
An Italian police source confirmed to the BBC that Zaghba, who lived in east London, had been on a watch list shared with many countries including the UK.
In March 2016, Italian officers stopped Zaghba at Bologna airport and found IS-related materials on his mobile phone. He was then stopped from continuing his journey to Istanbul.
Redouane was a chef who also used the name Rachid Elkhdar and police said he claimed to be Moroccan-Libyan. He married a British woman in Dublin in 2012 and lived in Rathmines in the Irish capital.
Butt featured in a Channel 4 documentary last year about Islamist extremists with links to the jailed preacher Anjem Choudary called The Jihadis Next Door.
The married father-of-two, who worked for London Underground as a trainee customer services assistant for nearly six months last year, could be seen in the programme arguing with police officers in the street, after displaying a flag used by IS in a London park.
Two people in Barking, east London, had also raised concerns about Butt, BBC home affairs correspondent Dominic Casciani said.
Metropolitan Police assistant commissioner Mark Rowley said an investigation into Butt began in 2015, but "there was no intelligence to suggest that this attack was being planned and the investigation had been prioritised accordingly".
At any one time there are about 500 active counter-terrorism investigations concerning 3000 people of interest.
So far four of those killed in the attack have been named.
Kirsty Boden worked as a senior staff nurse at Guy's and St Thomas' Hospital in London. Her family described her in a statement as an "outgoing, kind and generous person".
"We are so proud of Kirsty's brave actions which demonstrate how selfless, caring and heroic she was, not only on that night, but throughout all of her life," they added.
The hospital said Ms Boden was "an outstanding nurse and a hugely valued member of the staff team in Theatres Recovery, described by her colleagues as 'one in a million' who always went the extra mile for the patients in her care".
Canadian national Chrissy Archibald, 30, was the first victim to be named. Her family said she had died in her fiancé's arms after being struck by the attackers' speeding van.
A French national was also killed in the attack, according to foreign minister Jean-Yves Le Drian, which the BBC understands to be Mr Pigeard.
Boro Bistro manager Vincent Le Berre told Brittany news outlet Le Telegramme how his colleague was attacked in a bar near Borough Market.
"I managed to escape him, but my friend Alexandre did not have that chance," he said. "He was hit in the neck with a knife."
The family of 32-year-old James McMullan, from Hackney, east London, said they believed he was also dead.
Mr McMullan's sister said his bank card was found on a body at the scene.