Necessity of the Cartan subalgebra assumption in the distortion estimate (Proposition 3.*)

Ascertain whether the Cartan subalgebra assumption on the generating element X is necessary for the polynomial distortion conclusion in Proposition 3.* on evaluating distortion in completely solvable Lie groups; specifically, determine if the result remains valid without assuming that X lies in a Cartan subalgebra.

Background

Proposition 3.* provides a distortion estimate for one-parameter subgroups in completely solvable Lie groups: when the generator X lies in a Cartan subalgebra and has finite depth in the descending central series, the distortion is polynomial of degree equal to that depth; if the depth is infinite, the distortion is exponential.

Immediately after the proof, the authors note uncertainty about whether the Cartan subalgebra assumption is actually required for the polynomial distortion conclusion, highlighting a gap in the current understanding of distortion behavior absent Cartan regularity.