Given a free Lie algebra $L$ over a field $K$ with $k$ generators $x_1,\ldots, x_k$, is it true that the $K$-span of the set
$$\{[x_i, v] : 1\leq i \leq k, v\in L\}$$
is the entire Lie algebra $L$? If so, how could one prove this? If not, can we obtain something similar, e.g., if we assign weight one to every generator, it is true for odd weight elements?
Let $Y = \{[x_i,v] | v \in L, 1 \leq i \leq k \}$. Since $L$ is free, $x_1, \dots, x_n$ are not in $Y$, so the answer is negative.
On the other hand, as a vector space $L = X \oplus Y$ where $X$ is the $K$-span of the generators. So $Y$ generates "almost everything".