If one wanted to obtain a fan for a toric variety of dimension $ n>1 $ whose Cox ring is $ mathbb{Z}^{2} $ graded with weights $ {(a_{i},b_{i})}_{i=1}^{n+2} $, then one could let $ B $ be the $ (n+2)times 2 $ matrix whose $ i $-th row has entries $ (a_{i},b_{i}) $. After computing the nullspace, one ends up with an $ n times (n+2) $ matrix $ A $ with entries in $ mathbb{Z} $. The $ n+2 $ columns of $ A $ are the rays for a fan in $ mathbb{R}^{n} $. If the rays are $ {u_{rho_{1}},dots,u_{rho_{n+2}} } $, then maximal dimensional cones $ sigma $ are of the form $ operatorname{Cone}(u_{rho_{i_{1}}},dots,u_{rho_{i_{n}}}) $. The fan $ Sigma $ is then obtained from the maximal cones and their faces. From $ Sigma $ one obtains an ideal $ B(Sigma) = langle x^{widehat{sigma}} rangle_{sigma in Sigma} $ where $ x^{widehat{sigma}} $ is $ prod_{i mid rho_{i} notin sigma} x_{i} $. From here the quotient of $ mathbb{A}^{n+2}_{mathbb{C}} setminus Z(B(Sigma)) $ by the $ mathbb{G}_{m}^{2} $ action which sends $ x_{i} $ to $ z_{1}^{a_{i}}z_{2}^{b_{i}}x_{i} $ is isomorphic to the variety $ X_{Sigma} $ obtained from the fan $ Sigma $. As a result, the Cox ring of $ X_{Sigma} $ has the desired grading.

What if instead of wanting to find an explicit fan of a toric variety of dimension $ n>1 $ whose Cox ring is $ mathbb{Z}^{2} $ graded, one wants to find an explicit fan of a toric variety of dimension $ n>1 $ whose Cox ring is $ operatorname{Hom}(mathbb{Z}/langle M rangle mathbb{Z}, mathbb{C}^{ast}) times operatorname{Hom}(mathbb{Z}/langle N rangle mathbb{Z}, mathbb{C}^{ast}) $ graded with weights $ (overline{a_{i}}, overline{b_{i}})_{i=1}^{n} $? Is there a similar algorithm for obtaining the fan for such a variety?