The question is about the family of tensors that are naturally associated to any nice Lie group.
Take the Mauer-Cartan form, $\omega=g^{-1} dg$ and I would like to make the covariant index of this one-form explicit, $\omega\_a =g^{-1}\partial\_a g$. Then the Riemannian, left/right invariant, metric is just $g\_{ab}=tr(\omega\_a\omega\_b)$. More generally one can construct a family of left-right invariant tensors $T\_{a\_1...a\_s}=tr(\omega\_{a\_1}...\omega\_{a\_s})$. Does this family have any meaning/application/nice properties?



tensor hierarchy for Lie groups from Maurer-Cartan form
Sounds like you are in physics; particle physicists often assume that all Lie groups are compact. Compact Lie groups admit biinvariant metrics. The tensors you are considering are essentially the characteristic polynomials in the adjoint representation, so if you insert wedge product signs (work with the associated alternating tensors) then they represent certain of the Chern-Weil invariant differential forms that give rise to characteristic classes, if I understand your notation correctly. There may be some other Chern-Weil forms (like the Pfaffian, if $G=SO(2n)$).


