The Piper Seneca turboprop was flying from the eastern region of Papua to Tual city in the Maluku Islands when it was caught in a storm around midday.
Witness Akib Hanubun told AFP the aircraft was pounded by fierce winds and rain before being hit by lightning and crashing near a village.
"The plane's left wing broke after it was struck by lightning," the 47-year-old told AFP.
"Then the plane appeared to tilt to one side before crashing to the ground loudly and bursting into flames."
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Transport ministry spokesman Bambang Ervan confirmed that all four people on board the aircraft were killed instantly.
"They were the pilot, two engineers and a ground handler," he said, adding he had no further details about the crash at this stage.
The Piper Seneca is produced by US-based Piper Aircraft. The jet that crashed was operated by Intan Angkasa Air Service, according to Ervan.
The company's website says it supports "the oil and gas, mining, survey, forestry and many other industries" across Indonesia.
In April last year a Lion Air passenger jet carrying 108 people crashed into the sea after missing the runway as it came in to land on the resort island of Bali. No one died but dozens were injured.