Normed Lie products form a generating set of graded component of free Lie algebra

Statement for left-normed Lie products
Suppose $$L$$ is a free Lie ring on $$n$$ variables $$x_1,x_2,\dots,x_n$$. Consider all the left-normed Lie products in $$L$$ of weight $$r$$ with inputs coming from the generating set, i.e., for any function $$f:\{1,2,\dots,r\} \to \{ 1,2,\dots,n\}$$ consider the product:

$$[[\dots [[x_{f(1)},x_{f(2)}],x_{f(3)}],\dots,x_{f(r-1)}],x_{f(r)}]$$

These Lie products form a generating set for the degree $$r$$ graded component of $$L$$.

Statement for right-normed Lie products
Suppose $$L$$ is a free Lie ring on $$n$$ variables $$x_1,x_2,\dots,x_n$$. Consider all the right-normed Lie products in $$L$$ of weight $$r$$ with inputs coming from the generating set, i.e., for any function $$f:\{1,2,\dots,r\} \to \{ 1,2,\dots,n\}$$ consider the product:

$$[x_{f(1)},[x_{f(2)},[x_{f(3)},\dots,[x_{f(r-1)},x_{f(r)}]\dots]]]$$

These Lie products form a generating set for the degree $$r$$ graded component of $$L$$.

Caution
Note first that we can arrange a bijection between the left-normed and right-normed Lie products by repeated use of skew-symmetry, so we can restrict attention to right-normed Lie products.

The right-normed Lie products form a generating set for the graded component, but they do not form a freely generating set. In particular, there may be relations between the right-normed Lie products. First of all, note that right-normed Lie products where the last two inputs are equal are zero, hence can be dropped. Second, note that by skew-symmetry, the last two inputs can be interchanged, so we can remove one of ever pair of right-normed Lie products that are equivalent in this manner. This still leaves a total of $$(n^r - n^{r-1})/2$$ right-normed Lie products.

There may, however, be further relations between these Lie products due to the Jacobi identity. For instance, we can use the Jacobi identity in $$x,y,[x,y]$$ to obtain:

$$[x,[y,[x,y]] = [y,[x,[x,y]]$$

Thus, although there are four right-normed Lie products up to skew symmetry in the last two variables, the rank of the free abelian group is only three.

Related facts

 * Basic Lie products form a freely generating set of graded component of free Lie algebra: If we use basic Lie products instead of normed Lie products, we can obtain a freely generating set.
 * Normed Lie products need not form a basis for graded component of free Lie algebra