Lie algebras of CL type (Q1614628)

Finite-dimensional simple Lie algebras over an algebraically closed field of characteristic \(p > 7\) are categorized as being of either classical or Cartan type. Additionally, in smaller characteristics there exist simple Lie algebras that do not occur for characteristic \(p > 7\). Yet their structures have close affinities to algebras of either classical or Cartan type. This paper proposes to define these two types in such a way that the finite-dimensional simple Lie algebras of all characteristics can be classified as being of one or the other of these types. The author defines an element \(x\) of a Lie algebra \(L\) to be strongly nilpotent if ad \(x\) is nilpotent and also ad \(y\) is nilpotent for all \(y \in \text{ad}\;x \cdot L\). A simple Lie algebra is said to be of CL type if it has no nonzero strongly nilpotent elements. Otherwise it is said to be of CA type. A number of examples of simple Lie algebras of characteristics 2 and 3 are shown to be of CL type.