The question is about the family of tensors that are naturally associated to any nice Lie group. Take the MauerCartan form, $\omega=g^{1} dg$ and I would like to make the covariant index of this oneform 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 leftright invariant tensors $T_{a_1...a_s}=tr(\omega_{a_1}...\omega_{a_s})$. Does this family have any meaning/application/nice properties?

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 ChernWeil invariant differential forms that give rise to characteristic classes, if I understand your notation correctly. There may be some other ChernWeil forms (like the Pfaffian, if $G=SO(2n)$). 


Expanding on Ben's answer: If you symmetrize, you get the left (or bi) invariant extension of the characteristic polynomials. Example: Let $G=SO(n,\mathbb R)$ be a simple matrix group. Then you get (symmetrized version) the Newton polynomials in the eigenvalues $\sum \lambda_i^p$ of $g^{1} X$ if $(g,X)$ is a tangent vector with foot point $X$ of $SO(n)$ in $Mat(n)$. 

