New methods for the classification of the simple modular Lie algebras

We investigate the structure of simple modular Lie algebras $L$ over an algebraically closed field of characteristic $p>7$. Let $T$ denote an optimal torus in some $p$-envelope $L_p$. We prove: If $Q(L,T)=L$ and $C_L(T)$ is a Cartan subalgebra, then $L$ is classical. If $Q(L,T)\ne L$ and $C_L(T)$ distinguishes the roots of $T$ on $L/Q(L,T)\ne 0$, then $L$ is of Cartan type.
The methods give new proofs even for the restricted simple Lie algebras.