16.15 (2006)

An element $g$ of a group $G$ is an Engel element if for every $h \in G$ there exists $k$ such that $[h, g, \dots, g] = 1$, where $g$ occurs $k$ times; if there is such $k$ independent of $h$, then $g$ is said to be boundedly Engel.
$\qquad$ a) (B. I. Plotkin). Does the set of boundedly Engel elements of a group form a subgroup?
$\qquad$ b) Does the set of boundedly Engel elements form a subgroup in a torsion-free group?
$\qquad$ c) The same question for right-ordered groups?
$\qquad$ d) The same question for linearly ordered groups?