The plane was in the air for a short time before crashing into the hangar at Parafield Airport, north of Adelaide, shortly after 2pm on Wednesday.
Ten people inside the hangar were injured, including one person who was left with life-threatening burns, while two others remain in serious conditions.
They were rushed to Royal Adelaide Hospital for treatment, along with a fourth person who suffered minor injuries.
Another six people were treated for smoke inhalation at Lyell McEwin Hospital and Modbury Hospital.
Aviation investigators spent Thursday morning combing through wreckage at the hangar, which was holding several other aircraft as well as a significant amount of fuel.
Australian Transport Safety Bureau chief Angus Mitchell said the training plane had been doing a circuit in the lead-up to the crash.
"We do know that quite tragically the instructor and the student have lost their lives," he told ABC TV.
He said investigators were looking into the nature of the flight, its take off, and whether it was part of "structured training".
"We're yet to find that out," Mr Mitchell said.
The hangar was being used for maintenance, aircraft storage and included a classroom space.
Metropolitan Fire Service chief officer Jeff Swann said impact from the crash had knocked out the hangar's internal sprinkler system.
On arrival, he said firefighters were faced with "really thick, black smoke and intense flames".
"These flames were further intensified by a significant amount of fuel inside the hangar, and also the fact that the aircraft had impacted the fire hydrant system," Mr Swann said.
The ATSB investigation has been bolstered by experts from Canberra and Brisbane, who arrived at the scene on Thursday morning.
They are undertaking site and wreckage examination, and recovering aircraft components for examination at technical facilities in Canberra.
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