Following Neeman's article "New axioms for triangulated categories", for a triangulated category $\mathscr T$ let $CT(\mathscr T)$ denote the category of candidate triangles, i.e. diagrams \begin{equation}X\overset f\to Y\overset g\to Z\overset h\to \Sigma X\quad (*)\end{equation} such that $gf=0$, $hg=0$ and $(\Sigma f)h=0$, with morphisms being commutative diagrams between such triangles. We can define homotopy of maps between candidate triangles to be chain homotopy and there is an automorphism $\tilde\Sigma\colon CT(\mathscr T)\to CT(\mathscr T)$ which takes $(*)$ to $$Y\overset{-g}\to Z\overset{-h}\to \Sigma X\overset{-\Sigma f}\to \Sigma Y.$$ We can define mapping cones as in a usual chain complex category, and a lot of the usual results hold for this category (e.g. homotopic maps have isomorphic mapping cones).
What I fail to see, is why the mapping cone construction along with $\tilde\Sigma$ does not give rise to a triangulation of $CT(\mathscr T)$?