Characterization of free Lie ring in terms of eigenspaces of Dynkin operator

Statement in eigenspace terms
Suppose $$A = \mathbb{Q}\langle x_1,x_2,\dots,x_n \rangle$$ is a free associative $$\mathbb{Q}$$-algebra on finitely many variables $$x_1,x_2,\dots,x_n$$. Note that this is like a polynomial ring, except that the variables do not commute. Let $$\mathbb{Q}L$$ be the $$\mathbb{Q}$$-Lie subalgebra of $$A$$ generated by $$x_1,x_2,\dots,x_n$$. It is true that $$L$$ is the free Lie algebra on $$x_1,x_2,\dots,x_n$$.

Let $$k$$ be a positive integer. The claim is that the homogeneous degree $$k$$ component of $$\mathbb{Q}L$$ is equal to the eigenspace of the Dynkin operator with eigenvalue $$k$$. In other words, an element $$a \in A$$ is in this degree $$k$$ component if and only if $$\delta(a) = ka$$, where $$\delta$$ denotes the Dynkin operator.

Note that the statement holds independent of whether we choose the left-normed or right-normed Dynkin operator.

In particular, since $$\mathbb{Q}L$$ is itself the direct sum of its homogeneous degree components, we obtain that $$\mathbb{Q}L$$ is the direct sum of its eigenspaces for all positive integers.

Statement in terms of ideals in the symmetric group
Note that the Dynkin operator can be thought of as multiplication by the following element of $$\mathbb{Q}[S_k]$$ (where we denote the identity element by $$1$$:

$$(1 - (1,2,3,\dots,k)) \dots (1 - (1,2,3))(1 - (1,2))$$

With this interpretation, the result says that for a given $$k$$, the Schur functor describing the degree $$k$$ Lie operad is the left ideal generated by the Dynkin element.

Further, this tells us that the image of the degree $$k$$ component of the associative algebra under the Dynkin element gives us the degree $$k$$ component of $$\mathbb{Q}L$$.

Significance for converting between formulas
The above characterization is significant in the following sense. Given a formula for an element of $$\mathbb{Q}L$$ in terms of sums of associative products, we can convert the formula to a formula in terms of sums of Lie products as follows.

Left-normed Lie product version

 * Replace each associative product by the corresponding left-normed Lie product.
 * Also, insert an extra factor in the denominator corresponding to the degree (weight) of the associative product.

Right-normed Lie product version

 * Replace each associative product by the corresponding right-normed Lie product.
 * Also, insert an extra factor in the denominator corresponding to the degree (weight) of the associative product.