Isometries in the diamond

We show the (anti)holomorphicity of smooth Kobayashi isometries of the diamond, the domain defined as $\mathrm{△}:=\{z\in {\mathbb{C}}^2:\mathrm{\mid }{z}_1\mathrm{\mid }+\mathrm{\mid }{z}_2\mathrm{\mid }<1\}$. Additionally, we discuss the problem of uniqueness of real geodesics, left inverses and strict convexity of indicatrices.