The full automorphism group of $\bar{T}$

Let $\bar{G}$ be the wonderful compactification of a simple affine algebraic group $G$ of adjoint type defined over $\mathbb{C}\mathrm{.}$ Let ${\overline T}\subset \overline G$betheclosureofamaximaltorus$T\subset G.$ We prove that the group of all automorphisms of the variety $\bar{T}$ is the semi-direct product $N_G(T)⋊D,$ where $N_G(T)$ is the normalizer of $T$ in $G$ and $D$ is the group of all automorphisms of the Dynkin diagram, if $G≠\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{C})$. Note that if $G=\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{C})$, then $\bar{T}=\mathbb{C}{\mathbb{P}}^1$ and so in this case $\text{Aut}(\bar{T})=\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{C})$.