Given a Lie algebra (finite-dimensional, over a field) and a basis, denote the structure constants in the usual way: $[e_i,e_j]=\sum_kc_{ij}^ke_k$. We say that the structure constants are cyclic if $c_{ij}^k=c_{jk}^i$ for all $i,j,k$ (note that this depends on the choice of basis).

The Lie algebra being given, the existence of a basis with cyclic structure constant is a nontrivial condition as it can easily be checked not to exist in a non-abelian 2-dimensional Lie algebra.

Which Lie algebras admit a basis with cyclic structure constants?