I saw two definition of Deligne torus: one is $\mathbb{S}=\mathrm{Res}^{\mathbb{C}}_{\mathbb{R}}G_m$, another is as a closed subgroup scheme of $GL(2)_{/\mathbb{R}}$ defined by $\mathbb{S}=\{g\in GL(2)_{/\mathbb{R}}\mid g=\begin{pmatrix} a & b \newline-b & a \end{pmatrix}\}$ as a funtor of points. I think they should be equivalent, but $\mathbb{S}(\mathbb{C})=\mathrm{Hom}_{\mathbb{R}}(\mathbb{C}[t,t^{-1}],\mathbb{C})=(\mathbb{C}\otimes_{\mathbb{R}}\mathbb{C})^*$ by the first definition, whereas by the second definition $\mathbb{S}(\mathbb{C})=\{\begin{pmatrix} a & b \newline-b & a \end{pmatrix}\mid a,b\in \mathbb{C}, a^2+b^2\ne 0\}$, which is isomorphic to $\mathbb{C}^*\times\mathbb{C}^*$ by $\begin{pmatrix} a & b \newline-b & a \end{pmatrix}\to (a+bi,a-bi)$, is there anything wrong?
Any help will be appreciated, and this question comes from my reading of Peters and Steenbrink's book Mixed Hodge Structures P.36.
