ℤ × ℤ‐Graded Lie Algebras Generated by the Virasoro Algebra and sl 2

The present paper is another step toward the classification of ℤ × ℤ‐graded Lie algebras: we classify ℤ × ℤ‐graded Lie algebras = ⊕ i,j ∈ℤ i,j over a field of characteristic 0 with dim i,j ≤ 1 for i, j ∈ ℤ, satisfying: (1) 0 = ⊕ i ∈ℤ i ,0 ≃ Vir, the centerless Virasoro algebra, (2) 0,–1 , 0,0 , 0,1 span the 3‐dimensional simple Lie algebra sl 2 , (3) is generated by the centerless Virasoro algebra and sl 2 . These algebras include some rank 2 centerless Virasoro algebras, and some rank 2 Block algebras and their extensions.