Mr Carter's friend Jonathan Waller, and Stephen Holdich, the co-owner of Atlas helicopters, are also reported to have been onboard.
A statement was released with "aching sorrow" by the Carter family, which said that Phillip and Andrew were "much loved and cherished family members".
"It is difficult to find words to describe our feelings over the tragic loss of two people whom we all loved so much," the family added.
Businessman Mr Carter founded the vocational qualifications firm Carter & Carter and was a wealthy Chelsea supporter.
The company said last night that he was an "inspirational leader".
"Above all Phil was a great friend, a strong family man, and will be deeply and sadly missed by all those who knew him," a statement added.
A search operation was launched by police yesterday after the helicopter failed to reach its planned destination at a private residence in Thornhaugh.
Its largely intact wreckage was discovered in Bedford Purlieus wood between Wansford and Duddington with the four bodies onboard.
They have since been taken to Peterborough District Hospital where post mortem examinations are being carried out.
Investigations are underway at the scene of the crash by the Air Accident Investigations Branch.
Click here to run an energy price comparison, and see if you could be paying less for your gas and electricity.