Solvabilizers of Lie algebras of Caratan type

Determine the solvabilizer sol(L) for finite-dimensional Lie algebras of Caratan type, thereby resolving the relationship between the solvabilizer and the solvable radical in this class.

Background

The paper defines sol(L) as the set of elements y such that the two-generated subalgebra generated by x and y is solvable for every x in L. In characteristic zero, a cited theorem establishes that sol(L) coincides with the solvable radical R(L), and an analogous equality is proved under positive-characteristic hypotheses when L/R(L) is classical semisimple.

The authors identify the determination of sol(L) for Lie algebras of Caratan type as unresolved. This problem concerns extending the structural characterization of the solvabilizer beyond the classes treated by the preceding results.