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. 



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> Which Lie algebras admit a basis with cyclic structure constants?
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Lie algebra with cyclic structure constants
Endow your algebra with the symmetric bilinear form making $\{e\_i\}$ an orthornormal basis (the Gram matrix is the identity matrix). Since $c\_{ij}^k=([e\_i,e\_j],e\_k)$, the cyclicity condition is equivalent to the invariance of the form:
$$
([e\_i,e\_j],e\_k)=(e\_i,[e\_j,e\_k]).
$$
Thus a finite-dimensional Lie algebra admits a basis with cyclic structure constants if and only if it admits an invariant symmetric bilinear form with Gram matrix the identity matrix. If the base field is algebraically closed not of characteristic 2, the latter condition simplifies to "admits a nondegenerate invariant symmetric bilinear form".


