Let $\mathfrak{g}$ be a complex semi simple Lie algebra. Then $\mathfrak{g}$ is equipped with a canonical $\mathfrak{g}$-invariant non-degenerate bilinear form $\beta$. Now this gives a $\mathfrak{g}$-invariant non-degenerate bilinear form $\beta_m$ on $\mathfrak{g}^{\otimes m}$.
On the other hand, every irreducible $\mathfrak{g}$-module admits a contraviariant non-degenerate bilinear form. If $\mathfrak{g}^{\otimes m}$ decomposes into a sum of irreducible summands, each equipped with that non-degenerate bilinear form, then the sum of those also gives rise to a non-degenerate bilinear form on $\mathfrak{g}^{\otimes m}$.
Is there any chance, that these two bilinear forms on $\mathfrak{g}^{\otimes m}$ are the same?