For a simple Lie algebra $\mathfrak{g}$ (over $\mathbb{C}$) and its adjoint group G, the $G$-invariant polynomials on $\mathfrak{g}$ are linear combinations of products of traces, i.e. functions of the form $$P(X) = tr_V(\pi(X)^n)$$ for representations $(V,\pi)$, by the Chevalley Restriction theorem.

Is this also true for $G$-invariant tensors? Are the elements of $\bigotimes^* \mathfrak{g}^\vee$ also linear combinations of tensor products of traces, i.e. functions of the form $$T(X_1,\ldots,X_n) = tr_V(\pi(X_1)\cdots\pi(X_n)),$$ along with rearrangement of the tensor indices?

Note: I am only interested in tensor invariants of the adjoint representation; the tensor invariants of the minimal representations have been worked out to various extents, but those are not the ones I'm directly interested in.

I have managed to prove this for the classical Lie algebras, and indeed for $A_r, B_r$ and $C_r$ we can take the minimal representation for $(V,\sigma)$. For $D_r$ we need to do something for the Pfaffian, but once we have that we have everything. All of this follows from the invariant theory of these Lie algebras, which is fairly straightforward.

Unfortunately, the invariant theory of the exceptional Lie algebras is somewhat more complicated, and so I haven't been able to show this statement for them. I think it's true, and I have some little bits of computational evidence that we only have to use the minimal representations for these cases as well, but no proof. Is there any literature on this?