Why the octahedral axiom?

My question is about the octahedral axiom (OA) in the definition of a triangulated category. For what I can understand so far (cf. Huybrechts, Fourier-Mukai in algebraic gometry, Definition 1.32), this axiom wants to roughly generalise the "double quotient" situation in the category of abelian groups, i.e. if $A\subset B\subset C$ are abelian groups then $C/B\cong(C/A)/(B/A)$.

I would like to know why people think that this axiom is superfluous.

I reported the "double quotient" situation because one may say that if it wants to generalise a situation which is natural in the non-generalised case, then one would expect this situation to be natural too. But it seems to me that this argument is too weak and probably there are better arguments...

Moreover, is it true that everyone is convinced about that?

As a motivation to this question I would like to say that: $1)$ last summer a paper by Maciocia appeared in which he proved the OA is a consequence of the previous ones. But unfortunately there was an error which is still not fixed. $2)$ In the Huybrechts book, I have cited before, he doesn't state the OA because he "will never use it explicitely and only once implicitely". $3)$ I am very curious about that.

Thank you all!

P.S.: I have tried to find something in the literature or in StackExchange as well but I was apparently unable. Sorry if it is a duplicate.

Solution 1:

The octahedral axiom is probably not superfluous. That is, Maciocia's error will probably never be repaired. People were very surprised by his claim and unsurprised by the discovery of the flaw.