"We as the police have started an investigation, I mentioned when I was down at the scene this morning that one of our very senior investigating officers is leading that for us", the commander said.
London fire officials said Thursday firefighters had put out a blaze that killed 12 people as it raced through a 24-story apartment building a day earlier, and that an unknown number of people remained inside.
Asked about the number of dead, he said he hoped the death toll would not reach "triple figures".
He said the search of the gutted block might take months and some victims might never be identified.
37 victims are now being treated in hospital, of which 17 are in critical care.
"I spoke to one of my officers who was very near when someone came out the window, and he was in tears and he is a professional fire officer", she said.
Pressed over this, Mrs May said: "What we need to do is to ensure this awful tragedy is properly investigated". But the shock at its scale turned to anger and recriminations on Thursday.
The leader of Kensington and Chelsea Council - the authority that owns the tower block - told BBC Two's Newsnight it would not use the type of cladding fitted to Grenfell Tower in other buildings in the borough.
A tenants' group had complained for years about the risk of a fire in the building, and authorities have refused to speculate on what could have started the blaze.
"I anticipate that London Fire Brigade will be on scene here for many days to come, working with our partners to ensure that we do the best for the people who are waiting for news of their loved ones".
Eldest son Lee, 47, posted a Facebook message reading: "If anyone has seen my dad, Tony Disson, could they let us know".
He added: "I called them back but they were talking to emergency people on their phone. and that's the last I heard from any of them".
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But in another show of bipartisanship, Thursday night's congressional baseball game will go ahead as scheduled, in Nationals Park. Democratic former Alexandria Mayor Bill Euille said he had spoken often with the man on recent mornings at the nearby YMCA.
The building was recently subject to a multi-million dollar renovation, and local councilors told CNN modern safety certifications had been issued.
The firm behind the work said the project met all required building regulations.
Ahmed Chellat said his sister and some friends had escaped the blaze but five of his extended family were still missing: his brother-in-law Aziz el Wahabi, Wahabi's wife Fouzia and their three children aged 20, 15 and 9.
"We've only brought water for now, but the plan is to get our 24/7 breakfast operation up and running as soon as we can", said graphics designer Gobind Singh, 33, the Daily Mail reported.
More than 1 million pounds ($1.27 million) has been raised to help victims of the tragedy as volunteers and charities worked through the night to find shelter and food for people who lost their homes.
Meanwhile, 44 households were given emergency accommodation.
Others gathered seeking news of the missing.
Resident Rania Ibrham streamed the video live on her Facebook page at 1:38 a.m. local time. She told the Guardian, the Grenfell disaster is "unforgivable." .
"They felt the fire quickly, and they saw the heat so they just ran out. Maybe 70 percent are from Somalia, Sudan, Morocco".
The star urged her followers to also help out if they can. The singer appeared at the Grenfell Tower to support the victims and survivors.
A look at the photos and video of the inferno that claimed at least 17 lives shows why the cladding is a possible culprit: The flames can be seen "climbing" up the sides of the public housing building at an alarming pace, overwhelming efforts to limit the destruction.
On Thursday morning, the first video emerged from inside the building, shot as it was engulfed in flames.
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