We use the tropicalization map to identify the canonical skeleton $\Delta(T)$ with $N_{\mathbb{R}}$. We denote by $C$ the open cone $(N_{\mathbb{R}}\times\mathbb{R}_{>0})\cup\{0\}$ in $N_{\mathbb{R}}\oplus\mathbb{R}$. Let $\Sigma$ be a locally finite fan of strongly convex rational polyhedral cones in $C$. We denote by $\Sigma_{1}$ the rational polyhedral complex in $N_{\mathbb{R}}$ obtained by intersecting the cones in $\Sigma$ with $N_{\mathbb{R}}\times\{1\}$. Consider the torus embedding $T\to\mathscr{X}$ over $R$ associated with $\Sigma$ as in [Kü98, 1.13]. The $R$-scheme $\mathscr{X}$ is separated and locally of finite type, and it is quasi-compact if and only if $\Sigma$ is finite. Since $\Sigma$ is supported in $C$, the generic fiber of $\mathscr{X}$ is canonically isomorphic to the split $K$-torus $T$. Assume that $\mathscr{X}$ is regular; this is equivalent to the property that the fan $\Sigma$ is simple, and it implies that the special fiber $\mathscr{X}_{k}$ is a strict normal crossings divisor. Denote by $\mathfrak{X}$ the formal $t$-adic completion of $\mathscr{X}$. The generic fiber $\mathfrak{X}_{\eta}$ is a $K$-analytic space endowed with a natural injective morphism of $K$-analytic spaces $i:\mathfrak{X}_{\eta}\to T^{\mathrm{an}}$. The morphism $i$ embeds $\mathfrak{X}_{\eta}$ as an analytic domain in $T^{\mathrm{an}}$.

The construction of the Berkovich skeleton $\mathrm{Sk}(\mathscr{X})$ and the retraction map $\rho_{\mathscr{X}}$ in [MN15, §3] are local on $\mathscr{X}$, so that they extend immediately to schemes that are locally of finite type. This yields a canonical embedding of the dual intersection complex $\Delta(\mathscr{X})$ of $\mathscr{X}_{k}$ into $\mathfrak{X}_{\eta}$. The image of this embedding is called the Berkovich skeleton of $\mathscr{X}$. The embedding has a canonical retraction $\rho_{\mathscr{X}}:\mathfrak{X}_{\eta}\to\Delta(\mathscr{X})$. It follows directly from the definitions that $\Delta(\mathscr{X})$ is contained in $\Delta(T)=N_{\mathbb{R}}$ and coincides with the support of $\Sigma_{1}$. In particular, if $\Sigma$ is a subdivision of $C$, then $\Delta(\mathscr{X})=\Delta(T)$. Moreover, the $\Delta$-structure on $\Delta(\mathscr{X})$ is precisely the polyhedral decomposition $\Sigma_{1}$. We have $\mathfrak{X}_{\eta}=\rho_{T}^{-1}(|\Sigma_{1}|)$, and the retraction map $\rho_{\mathscr{X}}$ is the restriction of $\rho_{T}$ to $\mathfrak{X}_{\eta}$.