Practically speaking, you can bring anything aboard a private jet that the pilot will allow. In most locations, there is no baggage screening or anything like that at the facilities (terminals) that serve general aviation aircraft.
Keep in mind that a lot of those prohibited items are prohibited because they are potentially unsafe. If you’re going to be transporting volatile chemicals or anything like that, please let your pilot know and please use the proper precautions for packing and storing such items.
The TSA’s position — so far — is that while airline passengers can hurt other people in the main terminal and others on their airplane, passengers on private jets can only hurt themselves and those in their party… so if they want to kill themselves, then let them do it.
As a sometimes-charter pilot myself, I ask you please not try to kill me if you’re riding on my airplane. If you’re unsure whether something is safe or not, ask your pilot and he will find out for you. Pressure and temperature changes (sometimes rapid) can cause some things to happen that you may not expect. You don’t want to find out the hard way…
On a lighter note, flying private does mean that you can bring all of your liquids on the airplane with you, and you can even drink your own alcohol if the pilot permits it. However, please note that the pilot is still prohibited by law from transporting intoxicated persons… so please don’t get drunk on the plane and get your pilot in trouble.
Firearms and other personal weapons may be carried on the plane if the pilot allows it and if airport police/security allows such weapons on the airport property of both your departure and arrival airports. Arrangements can usually be made for transport of such items if you let your pilot know in advance, so that he can confirm the proper procedures with airport police/security.