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Police say 32-year-old Emad Al Swealmeen, who was killed, boarded the taxi in the northern English city and asked to be driven to the hospital, which was about 10 minutes away.
The blast took place within the car as it approached the drop-off point in front of Liverpool Women's Hospital. The taxi driver was treated for injuries and released after medical attention.
The country has raised its national threat level to severe, with interior minister Priti Patel saying the blast in the northern English city was the second fatal incident in one month following the stabbing to death of lawmaker David Amess.
The explosion took just before 11am on Sunday (local time), when a Remembrance Day service to commemorate war dead was being held at nearby Liverpool Cathedral. Police are investigating whether the events are connected.
Police are searching two places associated with Al Swealmeen and have recovered significant items, Detective Chief Inspector Andrew Meeks said in a statement released by Greater Manchester Police.
"Now that we have released his name, any information that the public may have about Al Swealmeen, no matter how small, may be of great assistance to us," he said.
Police carried out a controlled explosion earlier on Monday as a precaution at Sefton Park.
Three men aged 29, 26 and 21 were arrested on Sunday and on Monday Assistant Chief Constable Russ Jackson of Counter-Terrorism Policing Northwest said another man, aged 20, had been arrested.
Jackson said it was unclear why the passenger had wanted to go to the hospital or what had caused the sudden explosion of the device.
Earlier, Liverpool Mayor Joanne Anderson described the actions of the taxi driver as heroic. The BBC reported that he had locked the passenger inside the vehicle.
"It is important that the public remains alert to the threat from terrorism but not alarmed," Patel said.