Note that a) I guess cyclicality $c_{ij}^k=c_{jk}^i$ is not base-independent, but it suffices (for me) that it holds in some basis, b) this caveat still leaves Lie algebras like aff1 ( $[a_1,a_2]=a_1$, so we're stuck with $[a_1,a_1]=0\neq a_2)$ where this can't hold in any basis. ($c_{ijk}$ is still cyclic in this case...but only because it vanishes identically.)
"$c_{ij}^k*c_{ik}^j$ is invertible" defines the semisimple Lie algebras - so would cyclicality too say us something about a Lie algebra?
Cf. Nilpotency of Lie Algebra from Structure Constants - I would too be interested which properties of a Lie algebra are easily read off from the $c_{ij}^k$.

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?

Note that a) I guess cyclicality $c_{ij}^k=c_{jk}^i$ is not base-independent, but it suffices (for me) that it holds in some basis, b) this caveat still leaves Lie algebras like aff1 ( $[a_1,a_2]=a_1$, so we're stuck with $[a_1,a_1]=0\neq a_2)$ where this can't hold in any basis. ($c_{ijk}$ is still cyclic in this case...but only because it vanishes identically.)
"$c_{ij}^k=c_{ik}^j$ is invertible" defines the semisimple Lie algebras - so would cyclicality too say us something about a Lie algebra?
Cf. Nilpotency of Lie Algebra from Structure Constants - I would too be interested which properties of a Lie algebra are easily read off from the $c_{ij}^k$.

Note that a) I guess cyclicality $c_{ij}^k=c_{jk}^i$ is not base-independent, but it suffices (for me) that it holds in some basis, b) this caveat still leaves Lie algebras like aff1 ( $[a_1,a_2]=a_1$, so we're stuck with $[a_1,a_1]=0\neq a_2)$ where this can't hold in any basis. ($c_{ijk}$ is still cyclic in this case...but only because it vanishes identically.)
"$c_{ij}^k*c_{ik}^j$ is invertible" defines the semisimple Lie algebras - so would cyclicality too say us something about a Lie algebra?
Cf. Nilpotency of Lie Algebra from Structure Constants - I would too be interested which properties of a Lie algebra are easily read off from the $c_{ij}^k$.