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Appendix A Notation

Symbol Description Location
\(\lielayer{\mathfrak{g}}{k}\) The homogeneous component of degree \(k\) of a stratified Lie algebra \(\mathfrak{g}\) Paragraph
\(L_{g}\) The left translation by a group element \(g\) Paragraph
\(\abnormalpolynomial{X}{\covector}\) The abnormal polynomial determined by the covector \(\covector\in\mathfrak{g}^*\) and vector \(X\in\mathfrak{g}\) Definition 2.1
\(\hallset\) A Hall set Definition 2.6
\(h\) A Hall tree Definition 2.6
\(w\) A Hall word Paragraph
\(\freelie{r}\) The free Lie algebra of rank \(r\) Subsection 2.3
\(\polyring{\hallset}\) The weighted multivariate polynomial ring of a free Lie algebra with respect to the Hall set \(\hallset\) Definition 2.13
\(XP\) The derivative of a polynomial \(P\) by a Lie algebra element \(X\) Definition 2.15
\(\covector_{w}\) The component of a covector \(\covector\) related to a Hall word \(w\) Lemma 2.17
\(\freelie{r,s}\) The free nilpotent Lie algebra of rank \(r\) and step \(s\) Lemma 3.1
\(\freecarnot{r,s}\) The free Carnot group of rank \(r\) and step \(s\) Lemma 3.1
\(\lowercentralseriesterm{\mathfrak{g}}{m}\) The \(m\text{:}\)th term of the lower central series of \(\mathfrak{g}\) Remark 3.3
\(\alpha\star\beta\) The concatenation of two curves \(\alpha\) and \(\beta\) Subsection 4.2