Solvability of finite-field S-Lie algebras in characteristic greater than two

Determine whether every finite-dimensional S-Lie algebra over a finite field of characteristic p>2 is solvable.

Background

An S-Lie algebra is defined by the condition that sol_L(x) is a Lie subalgebra for every x in L. The paper proves that over an algebraically closed field of characteristic zero, a finite-dimensional Lie algebra is solvable if and only if it is an S-Lie algebra.

The authors exhibit a simple S-Lie algebra over the field F_2, showing that the characteristic-zero equivalence does not extend directly to all finite fields. They leave unresolved whether the stronger characteristic assumption p>2 forces finite-field S-Lie algebras to be solvable.

References

However, it remains an open question as to whether $S$-Lie algebras over a finite field of characteristic $p>2$ are always solvable.

— The solvable Graph of a finite-dimensional Lie Algebra  (2511.08290 - Towers et al., 11 Nov 2025) in Section 3, immediately after the example of a simple S-Lie algebra over the field F_2