I am interested in finding formulas for commutators of symmetrized monomials in a universal enveloping algebra. Let $C(x_1,\ldots, x_n)= (1/n!)\sum x_{\sigma(1)}\cdots x_{\sigma(n)}$ where the sum runs over all permutations, and $x_i \in L$ for some LIe algebra $L$. This is an element of $UL$.

Now, reasonable combinatorics shows that, in $UL$, we have,

$ [C(x_1,\ldots, x_n), l] = \sum_{i=1}^n C(\ldots,[x_i,l],\ldots) $

for $l\in L$.

I am looking for formulas for $[C(x_1,\ldots, x_n), C(y_1,\ldots, y_m)]$ in terms of symmetrized monomials and brackets. Even for $n=m=2$ the number of terms gets fairly large. If anyone knows where I can find such things I would be very grateful.