Strong divisibility of the aggregate solvabilizer sizes

Prove that for every finite-dimensional Lie algebra L over a finite field F, the order |L| divides the sum of the cardinalities of the elementwise solvabilizers, namely $\sum_{x\in L}|sol_L(x)|$, equivalently that the average $\frac{1}{|L|}\sum_{x\in L}|sol_L(x)|$ is an integer.

Background

For each element x of a finite-dimensional Lie algebra L, the solvabilizer sol_L(x) consists of the elements y for which the subalgebra generated by x and y is solvable. Earlier results establish several coset decompositions and divisibility properties for individual solvabilizers, including divisibility by the field size over finite fields and, under additional assumptions, divisibility by the size of the solvabilizer of L or the centralizer of x.

The paper proposes a stronger global divisibility statement for finite fields: the total number of pairs (x,y) for which the generated subalgebra is solvable should be divisible by |L|. The authors report exhaustive computations for sl_2(F_3), sl_2(F_5), and gl_2(F_3), all of which support the conjecture, but no general proof is given.

References

Conjecture 1.\label{strong-div} Let $L$ be a finite–dimensional Lie algebra over a finite field $F$. Then $|L|$ divides $\sum_{x\in L} |sol_L(x)|$. Equivalently, the average size $\frac{1}{|L|}\sum_{x\in L} |sol_L(x)|$ is an integer.

— The solvable Graph of a finite-dimensional Lie Algebra  (2511.08290 - Towers et al., 11 Nov 2025) in Section 3, after Remark following Proposition 2.13, Conjecture 1 (label strong-div)