lieBasis -- basis element corresponding to a Lyndon word in a Lie algebra

A word $l$ on the alphabet $\{1,\dots, d\}$ is a Lyndon word if it is strictly smaller, in lexicographic order, than all of its rotations. To any Lyndon word we can associate an iterated Lie bracketing $b(l)\in T(\mathbb{R}^d)$ defined iteratively as follows. If $l$ is a letter $i\in \{1,\dots, d\}$ we simply define $$ b(i) := e_i$$ where as ever $e_i$ is the $i-th$ vector in the canonical basis of $\mathbb{R}^d$. For the length of $l$ greater than 1 we define $$ b(I) := [b(I_1), b(I_2)]$$ where $I_1, I_2$ are such that their concatenation $I_1 I_2$ is $I$ and $I_2$ is the longest Lyndon word appearing as a proper right factor of $I$.

This method computes $b(l)$ for a given Lyndon word. To illustrate its usage, we replicate Example 4.9 of the reference paper.

The word can also be given as a List.

Varieties Of Signature Tensors (doi.org/10.1017/fms.2019.3)