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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.

bliipbluup
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  • I have tried to find such a formula, but in general the Jacobi identity applied to the basis vectors of a Hall basis is difficult to understand. I tried for $\nu=1,2,3,4$ and $n=1,2,3,4$. This post shows how many vectors are "eliminated" by the Jacobi identity. It involves the Moebius $\mu$-function. – Dietrich Burde Jul 27 '20 at 16:54
  • @DietrichBurde Thanks for your reply. The post you mentioned was the starting point that led to this question :) – bliipbluup Jul 28 '20 at 07:28
  • Is there any other basis for which the Jacobi identity is well-understood? – bliipbluup Jul 28 '20 at 07:34

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