I've been working through some of the early parts of Neeman's book on triangulated categories, and he mentions that he does not know of a pre-triangulated category that is not triangulated. Is this still an open question? Actually, let me be a bit more specific and break this into two parts:
- The usual axioms for a triangulated category are known to be redundant. Do we know of a list of independent axioms? (Peter May wrote something about this, and it's the best I've seen but it still doesn't give independence results for the new list of axioms.)
- Assuming it is known that, say, the octahedral axiom is independent from the others, what is an example of a pre-triangulated category that is not a triangulated category?
It appears that if there is such an example, it would have to be very artificial... all of the pre-triangulated categories appearing in nature are automatically triangulated. (I haven't read all of May's article but it looks like he explains how one usually goes about proving that the octahedral axiom holds for a given category, so perhaps this would give some clues for situations where this would fail.)