Linearly ordered, non CT-groups A linearly ordered group is a group $(G,\dot,1)$ together with a linear ordering $<$ on $G$ with $f<g \Longrightarrow (f h<g h$ and $h f < h g)$ for all $f,g,h \in G$.
A commutative-transitive group (or CT-group) is a group $(G,\dot,1)$ in which two elements which commute with a third, non-trivial one, commute together. Equivalently, this is a group in which centralizers are commutative.
I'm having a hard time finding linearly ordered groups whose underlying group is not a CT-group, but I see no reason why any linearly ordered group should be a CT-group. Are there ways to construct linearly ordered non CT-groups?

Such groups are often called "bi-orderable".
Torsion-free nilpotent groups are all bi-orderable.
But a non-abelian nilpotent group is never commutative-transitive.
Hence, every non-abelian torsion-free nilpotent group yields an example (bi-orderable, not commutative-transitive).
