The Norming Set of a Bilinear Form on ℝ² with the Octagonal Norm

An element $(x_1, \ldots, x_n)\in E^n$ is called norming point of $T\in {\mathcal L}(^n E)$ if
$\|x_1\|=\cdots=\|x_n\|=1$ and $|T(x_1, \ldots, x_n)|=\|T\|$,
where ${\mathcal L}(^n E)$ denotes the space of all continuous $n$-linear forms on $E$. For $T\in {\mathcal L}(^n E),$ we define
$\text{\rm Norm\,}(T) = \{(x_1, \ldots, x_n)\in E^n: (x_1, \ldots, x_n) \ \text{is a norming point of}\ T\}.$
Let $\mathbb{R}^2_{o(w)}$ denote $\mathbb{R}^2$ with the octagonal norm with weight $0 < w\neq 1$
$\|(x, y)\|_{o(w)}=\max\big\{|x|+w|y|, |y|+w|x|\big\}.$
We classify $\text{\rm Norm\,}(T)$ for every $T\in {\mathcal L}(^2 \mathbb{R}_{o(w)}^2)$ with weight $0 < w\neq 1$ in this paper.