Zero and split triangles are distinguished

TR1 supplies $X\stackrel{1}{\to }X\to 0\to X[1]$; applying TR2 gives the displayed zero triangle.

Distinguished triangles are closed under finite biproducts. Indeed, TR1 completes the biproduct of their first arrows to a distinguished triangle, and TR3 on the two inclusions gives a morphism from the biproduct triangle to this completion that is the identity on its first two objects. For every $W$, the usual TR3 factorization argument on rotations gives exactness of the representable Hom sequence of a distinguished triangle; the sequence of the biproduct triangle is the direct sum of the two exact sequences. The elementary five-lemma diagram chase therefore makes the induced map on third objects bijective after applying $\mathcal{T}(W,-)$ for every $W$, hence an isomorphism by Yoneda. Thus the biproduct triangle is isomorphic to the TR1 completion and is distinguished.

The displayed split triangle is the biproduct of the distinguished identity triangle $X\stackrel{1}{\to }X\to 0\to X[1]$ and the distinguished zero triangle $0\to Y\stackrel{1}{\to }Y\to 0$. Step 2.1 therefore makes it distinguished.