$G$ Lie group Definition 2.1
$\mathfrak{g}$ Lie algebra of a Lie group Definition 2.1
$V_1\oplus\dots\oplus V_s$ stratification of a Lie algebra Definition 2.1
$\delta_{\dilationfactor}$ dilation by a factor $\dilationfactor\in\RR$ Definition 2.2
$\norm{\cdot}_d$ projection norm associated with a homogeneous distance Definition 2.4
$L_g$ left-translation by a group element $g\in G$ Definition 2.9
$u$ control of a horizontal curve Definition 2.9
$\gamma_\dilationfactor$ a curve dilated and reparametrized by a factor $\dilationfactor>0$ Lemma 2.12
$\partial E(Y)$ family of subdifferentials of a function $E$ at a point $Y$ Definition 2.17