The problem of classifying the finite dimensional simple Lie algebras over fields of
characteristic p > 0 is a long standing one. Work on this question has been directed by the
Kostrikin Shafarevich Conjecture of 1966 which states that over an algebraically closed field
of characteristic p > 5 a finite dimensional restricted simple Lie algebra is classical or of
Cartan type. This conjecture was proved for p > 7 by Block and Wilson in 1988. The
generalization of the Kostrikin-Shafarevich Conjecture for the general case of not necessarily
restricted Lie algebras and p > 7 was announced in 1991 by Strade and Wilson and eventually
proved by Strade in 1998. The final Block-Wilson-Strade-Premet Classification Theorem is a
landmark result of modern mathematics and can be formulated as follows: Every simple finite
dimensional simple Lie algebra over an algebraically closed field of characteristic p > 3 is of
classical Cartan or Melikian type. > 3. The first volume contains the methods examples and a
first classification result. This second volume presents insight in the structure of tori of
Hamiltonian and Melikian algebras. Based on sandwich element methods due to A. I. Kostrikin and
A. A. Premet and the investigations of filtered and graded Lie algebras a complete proof for
the classification of absolute toral rank 2 simple Lie algebras over algebraically closed
fields of characteristic > 3 is given. Contents Tori in Hamiltonian and Melikian algebras
1-sections Sandwich elements and rigid tori Towards graded algebras The toral rank 2 case
