Thanks for the references! I'll take a look. I was actually hoping to extend some results on small representations, actually inspired by Broer's and Reeder's work, but that required knowing more about what the case of not-small representations looked like.

I meant to include products of such traces as well. So we include things such as $T(X_1,X_2,X_3)T(X_4,X_5,X_6,X_7,X_8)-T(X_2,X_3,X_4)T(X_5,X_6,X_7,X_8,X_1)+....$ which can be made to alternate. By rearrangement of indices, I tend to think of this in terms of Einstein notation, so I'm thinking $T_{a_1,a_2,...}$ and I want to be able to say that both $T_{a_1,a_2}T_{b_1,b_2,b_3}$ and $T_{a_1,b_1}T_{a_2,b_2,b_3}$ are tensor products of the same traces, differing by how the indices are assigned.