This is a curiosity question, so please don't ask for motivation.

Let $k$ be a commutative ring, and $L$ a $k$-module. The tensor algebra $\otimes L$ becomes a super-$k$-algebra by taking $\bigoplus\limits_{n\text{ even}}L^{\otimes n}$ as the even part and $\bigoplus\limits_{n\text{ odd}}L^{\otimes n}$ as the odd part. This canonically induces a supercommutator $\left[\cdot,\cdot \right]_{\mathrm{s}}$ on $\otimes L$ (which is simply

$\left[U,V\right]_{\mathrm{s}}=UV-\left(-1\right)^{nm}VU$

for any $U\in L^{\otimes n}$ and any $V\in L^{\otimes m}$).

Define a map $T:\otimes L\to \otimes L$ by

$T\left(u_1\otimes u_2\otimes ...\otimes u_k\right) = \sum\limits_{i=1}^{k} \left(-1\right)^i u_i \otimes u_1 \otimes u_2 \otimes ... \otimes u_{i-1} \otimes u_{i+1} \otimes ... \otimes u_k$

(this is for the pure tensors; for the rest, just continue this by linearity). It is easy to see that

$\left[\mathrm{Ker} T,\mathrm{Ker} T\right]_{\mathrm{s}}\subseteq \mathrm{Ker} T$

and that $\left[L,L\right]_{\mathrm{s}}\subseteq \mathrm{Ker} T$, so that by induction, any tree of supercommutator brackets decorated by elements of $L$ must lie in $\mathrm{Ker} T$. Is it all of $\mathrm{Ker} T$ or is there more?

On a related matter, how can I make a computer do this kind of algebra for me, say on $L^{\otimes 7}$ ? Are there Haskell/ML libraries for tensor algebras or is there CAS software for this? (I am aware that my questions are reducible to representation theory of $S_n$.) A link (preferrably with a short tutorial) would be very appreciated. Thanks!