Is there a general formula for the Jacobi identity on the free $\nu$-nilpotent Lie algebra $\mathfrak{FL}(\nu,n)$ on $n$ generators?
In math overflow I found a list of the Hall bases. Looking at Hall's paper, though, the proof only shows that the constructed generating system is a basis by showing every element has a unique representation as a linear combination.
Reutenauer's "Free Lie Algebras" -which was also mentioned in the link above- seems to do what I'm asking for. However, it's not accessible to me without first studying the entire theory on non-commutative polynomials.
Is there a general formula to find $a_X$ in, say, $[X_1,[X_2,[X_3,X_4]]]=\sum_{X\in H}a_XX$, where $H$ is the hall basis to the $4$-nilpotent free Lie algebra? Is there one for the general case of nilpotence $\nu$? I know $H$ can be computed here.