{{\mathcal O}}&{{\mathcal O}}\end{bmatrix}\otimes
\begin{bmatrix}
    {{\mathcal O}}&{{\mathcal O}}(-\lambda_3)\\
    {{\mathcal O}}&{{\mathcal O}}\end{bmatrix}\right)\end{aligned}$$ and similarly for the actions of ${\vec x}_2$ and ${\vec x}_3$. In this case $$P=
    \begin{bmatrix}
        {{\mathcal O}}\\{{\mathcal O}}\end{bmatrix}\otimes
    \begin{bmatrix}
        {{\mathcal O}}\\{{\mathcal O}}\end{bmatrix}\otimes
    \begin{bmatrix}
        {{\mathcal O}}\\{{\mathcal O}}\end{bmatrix}
    \; {\rm and}\;P({\vec x}_1)=
    \begin{bmatrix}
        {{\mathcal O}}\\{{\mathcal O}}(\lambda_1)
    \end{bmatrix}\otimes
    \begin{bmatrix}
        {{\mathcal O}}\\{{\mathcal O}}\end{bmatrix}\otimes
    \begin{bmatrix}
        {{\mathcal O}}\\{{\mathcal O}}\end{bmatrix}$$ and similarly for $P({\vec x}_2)$ and $P({\vec x}_3)$. The tilting bundle is $$T=P\oplus P({\vec x}_1)\oplus P({\vec x}_2)\oplus P({\vec x}_3)\oplus
P(1)$$ with endomorphism algebra given by $$\xymatrix{&P({\vec x}_1)\ar|{x_1}[dr]\\
P\ar|{x_1}[ur]\ar|{x_2}[r]\ar|{x_3}[dr]&P({\vec x}_2)\ar|{x_2}[r]&P(1)\\
&P({\vec x}_3)\ar|{x_3}[ur]}$$ We can always choose coordinates such that ${{\mathcal O}}(-\lambda_i)\hookrightarrow{{\mathcal O}}$ is given by $T_{i-1}=0$ for $i=1,2$ and ${{\mathcal O}}(-\lambda_3)\hookrightarrow{{\mathcal O}}$ is given by $T_1-T_2=0$ Thus the relation is $x_3^2=x_1^2-x_2^2$. The corresponding ${\mathbb{L}}$-graded ring is then $$\begin{aligned}
R&=k[T_0, T_1, X_1,X_2,X_3]/(X_1^2
-T_0, X_2^2-T_1, X_3^2-(T_0-T_1))\\
&\simeq k[X_1,X_2,X_3]/(X_3^2-(X_1^2-X_2^2)).\end{aligned}$$

Finally, we would like to present the simplest example possible on ${\mathbb{P}}^2_{T_0:T_1:T_2}$: one with $4$ weights, all equaling $2$. Here ${{\mathcal O}}={{\mathcal O}}_{{\mathbb{P}}^2}$ and we let $L_i$ be the hyperplane given by $T_{i-1}=0$ for $1\leq i\leq 3$ and let $L_4$ be given by $T_0+T_1+T_2=0$. Consider $$\Lambda=
    \begin{bmatrix}
        {{\mathcal O}}&{{\mathcal O}}(-L_1)\\
        {{\mathcal O}}&{{\mathcal O}}\end{bmatrix}\otimes
\begin{bmatrix}
        {{\mathcal O}}&{{\mathcal O}}(-L_2)\\
        {{\mathcal O}}&{{\mathcal O}}\end{bmatrix}\otimes
\begin{bmatrix}
        {{\mathcal O}}&{{\mathcal O}}(-L_3)\\
        {{\mathcal O}}&{{\mathcal O}}\end{bmatrix}\otimes
\begin{bmatrix}
        {{\mathcal O}}&{{\mathcal O}}(-L_4)\\
        