I asked a mixed-up version of this question earlier.

The Lie algebras I have in mind are the homotopy Lie algebras of wedges of finitely many spheres (in dimensions greater than $1$). Thus each element has a degree and the bracket is the Whitehead product, which satisfies a graded version of the Jacobi identity, etc. The homotopy Lie algebra of a wedge of spheres is free on generators $x_1, \ldots , x_n$ in bijective correspondence with the spheres in the wedge.

Inside of any (graded) Lie algebra $L$ we may form the span of the iterated brackets of the $x_i$ with no repeated factors -- call any element in this span "square-free". Note that Jacobi rewrites of square-free brackets are also square-free.

A sum of $i$-fold brackets is said to have weight $i$.

So let $L = L(x_1, \ldots, x_n, x_{n+1})$ be the free (graded) Lie algebra on the given generators, and let $L_i \subset L$ be the subalgebra $L(x_1, \ldots,\widehat{x_i}, \ldots x_n, x_{n+1})$. For each $i = 1, \ldots , n$, choose a square-free weight $n$ element $w_i \in L_i$, not all of which are zero. Is it true that

$$
\mathcal{Z} = \{ [w_i, x_i]\ | \ i = 1, \ldots, n\}
$$
is a linearly independent set?

I have verified this with MAPLE for $n\leq 4$, and I'd be happy even for an answer for the special case of ordinary Lie algebras. An answer to the general question would be helpful in resolving some questions about the Lusternik-Schnirelmann category of rational spaces.