CT and AC Lie algebras

Question

In group theory, the following two classes of groups are well-known.

1. AC-groups: a group is called an AC-group if the centralizer of every non-central element is abelian.
2. CA-groups (or CT-groups): a group is called a CA-group if the centralizer of every non-identity element is abelian.

For the Lie algebra side, a CT-Lie algebra is defined and studied in Ref.1. and Ref.2. (There might be some others references, but I couldn't find.)

A Lie algebra $L$ is said to be commutative transitive (CT for short) if for any non-zero $x, y, z \in L$, if $[x, y] = [y, z] = 0$, then $[x, z] = 0$. This type of Lie algebras can be compared to CA-groups.

The question is: Is a type of Lie algebras that is analogous to AC-groups?

I guess it should be defined as: A Lie algebra $L$ is said to be an AC-Lie algebra if for any non-central elements $x, y, z \in L$, if $[x, y] = [y, z] = 0$, then $[x, z] = 0$.

I couldn't find a reference for this.

References

1. I. Klep, P. Moravec, Lie algebras with abelian certralizers. Algebra Colloq., 4 (2010) 629--636
2. V. V. Gorbetsevich, Lie algebras with abelian centralizers, Mathematical Note, 101 (2017) No. 5 795--801.