Let $L$ be a finite-dimensional Lie algebra over a field $k$ of characteristic zero and $e\_1,\ldots, e\_n$ some basis of $L$. The formula $[e\_i,e\_j] = \sum\_k C\_{ij}^k e\_k$ determines the structure coefficients $C\_{ij}^k$. Given any ordered $k$-tuple $I = (i\_1,\ldots,i\_k)\in \lbrace 1,\ldots,n \rbrace^k$, define $e\_I = e\_{i\_1}\cdots e\_{i\_k}\subset U(L)$ and 
$$
e^S\_I = \frac{1}{n!}\sum\_{\sigma\in\Sigma(k)} e\_{\sigma(i\_1)}\cdots e\_{\sigma(i\_k)}\in U(L).
$$
As it is well known, from various forms of a PBW theorem, $e\_I$, for all $I$ with $i\_1\lt i\_2\lt \ldots \lt i\_k$, $k \geq 0$, form a basis and also $e^S\_I$ (for the same set of $I$-s) form a basis. I need explicit formulas for $e\_I$ in the linear basis of $e^S\_J$-s where the coefficients are expressed in terms of the structure constants $C\_{ij}^k$ (and combinatorial factors). In fact, for my present purposes, it would be enough to know explicitly the deepest, lowest order, linear term (but it is of course the hardest summand in the expansion). For example, for the easiest nontrivial case $k = 2$, 
$$e\_{(i,j)} = e\_i e\_j = \frac{1}{2}(e\_i e\_j + e\_j e\_i) + \frac{1}{2} \sum\_k C\_{ij}^k e\_k = e\_{(i,j)}^S + \frac{1}{2} \sum\_k C\_{ij}^k e\_k,$$
hence the linear term is $\frac{1}{2} \sum\_k C\_{ij}^k e\_k$.



monomials in the universal enveloping of a Lie algebra in terms of the symmetric basis
An explicit formula is given in [this paper](http://www.sciencedirect.com/science/article/pii/S0021980068800626) by L. Solomon. I copy the abstract here:



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> Let g be a Lie algebra over a field of characteristic zero. Let T be the tensor algebra of g, let S be the subspace of symmetric tensors and let J be the two-sided ideal of T generated by tensors x⊗y−y⊗x−[x, y]. One formulation of the P-B-W theorem states that T=S⊕J, direct sum. In this paper we give an explicit formula for the projection of T on S defined by this direct decomposition.
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