Z x Z-graded Lie algebras generated by the Virasoro algebra and sl₂

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The present paper is another step toward the classification of ×-graded Lie algebras: we classify ×-graded Lie algebras A=⊕ᵢ,ⱼ∈ᵢ,ⱼ over a field F of characteristic 0 with dim Aᵢ,ⱼ≤1 for i,j∈, satisfying: (1) Å₀=⊕ᵢ∈ Åᵢ,₀ is isomorphic to the centerless Virasoro algebra, (2) A₀,₋₁,A₀,₀,A₀,₁ span the 3-dimensional simple Lie algebra sl₂, (3) A is generated by the centerless Virasoro algebra and sl₂. These algebras include some rank 2 centerless Virasoro algebras, and some rank 2 Block algebras and their extensions.