The 27-year-old, who was arrested on suspicion of child neglect, will return to a south London police station in mid-January.
Scotland Yard said the investigation, which is being led by detectives from the South Area Command Unit, is continuing.
Emergency services were called to the terraced house at around 7pm on Thursday, and found a harrowing scene.
The London Fire Brigade said the first crews on scene were faced with an intense fire throughout the whole of the ground floor.
Brave firefighters in breathing apparatus entered the house and rescued the four young children.
They were given immediate CPR by emergency services at the scene, including paramedics.
All four children were rushed to two hospitals in south London where sadly they died.
Ch Supt Dave Stringer said in a statement: “The thoughts of everyone at the Metropolitan Police are with the family and friends of four young boys who lost their lives in this tragic incident.
“We know that it will have come as a great shock to the local community, and as part of that community we are here to offer whatever support we can.
“At 7pm last night [Thursday], we were called to reports of a fire in a residential property.
“Officers responded to the incident alongside colleagues from the London Fire Brigade and London Ambulance Service.
“There was nobody else inside the property at the time of the fire.”
A woman, 24, who claimed she was a friend of the family, said the two sets of boys were twins
She told LondonWorld: “The kids, the respect they have, the niceness they have.
“They were always all holding hands, walking up the high street, they would always do as they were told.”