On a remarkable class of left-symmetric algebras and its relationship with the class of Novikov algebras

We discuss locally simply transitive affine actions of Lie groups G on finite-dimensional vector spaces such that the commutator subgroup [G,G] is acting by translations. In other words, we consider left-symmetric algebras satisfying the identity [x,y].z=0. We derive some basic characterizations of such left-symmetric algebras and we highlight their relationships with the so-called Novikov algebras and derivation algebras.