  In this paper we study Lie 2-algebras over an algebraically closed field of
characteristic two, which have a triangulable Cartan subalgebra, and derive
some general properties of centerless ones. These properties allow us to do an
analysis on simple Lie 2-algebras of toral rank three and provide a necessary
condition for simplicity. By means of this latter condition we also conclude
that simple Lie 2-algebras with a triangulable Cartan subalgebra of toral rank
three and of dimension less than or equal to 16 cannot exist.
