# CT and AC Lie algebras

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.
• This depends on your definition of "analogous". For CT-groups you also only say "This type of Lie algebras can be compared to CA-groups." This is not very precise. Feb 12 at 15:18
• These definitions can be transferred immediately from groups to Lie algebra and vice-versa, you need no reference. By the way, commutative-transitive groups are widely used.
– YCor
Feb 13 at 21:00