Classifying the absolute toral rank two case

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The classification of finite dimensional simple Lie algebras over fields of characteristic p > 0 has been a longstanding challenge, influenced by the Kostrikin-Shafarevich Conjecture from 1966. This conjecture asserts that for an algebraically closed field with characteristic p > 5, a finite dimensional restricted simple Lie algebra is either classical or of Cartan type. Block and Wilson proved this for p > 7 in 1988. Strade and Wilson announced a generalization for non-restricted Lie algebras in 1991, which Strade proved in 1998. The Block-Wilson-Strade-Premet Classification Theorem is a significant milestone, stating that every simple finite dimensional Lie algebra over an algebraically closed field of characteristic p > 3 is of classical, Cartan, or Melikian type. This volume is the second of a three-volume series on classifying simple Lie algebras over algebraically closed fields of characteristic > 3. The first volume covers methods and initial classification results, while this second volume delves into the structure of tori in Hamiltonian and Melikian algebras. It provides a complete proof for classifying absolute toral rank 2 simple Lie algebras, utilizing sandwich element methods and insights from filtered and graded Lie algebras.