## AppendixANotation

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