$X_{{\mathbb{R}}}$. Inside $X$ we can consider the complexification of ${\mathrm{SE}_3}$, which we will denote by ${\mathrm{SE}_{3,{\mathbb{C}}}}$, hence we have an injective map $\Phi: {\mathrm{SE}_3}\hookrightarrow {\mathrm{SE}_{3,{\mathbb{C}}}}\subseteq X$.

Action of ${\mathrm{SE}_3}$ on $X$ {#compactification:action}
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In this subsection we want to extend the natural actions of ${\mathrm{SE}_3}$ on itself, given by composition on the left and on the right, to actions of ${\mathrm{SE}_3}$ on ${\mathbb{P}}^{16}_{{\mathbb{C}}}$ which restrict to actions on $X$. This will be useful in the proofs of Section \[bond\_geometric\], because it will allow us to exploit the symmetries of $X$.

Let $\sigma_1: v \mapsto M_1 v + y_1$ and $\sigma_2: v \mapsto M_2 v + y_2$ be isometries. Then the product $\sigma_{12} = \sigma_1 \sigma_2$ maps $$v \; \mapsto \; (M_1 M_2) v + (M_1 y_2 + y_1).$$ We set $M_{12} := M_1 M_2$ and $y_{12} := M_1 y_2 + y_1$. The remaining affine coordinates of $\sigma_{12}$ are $$\begin{spreadlines}{0.8em}
        \begin{aligned}
            x_{12} & = -M_{12}^t y_{12} = -M_2^t M_1^t M_1 y_2 - M_2^t M_1^t y_1 \\
            & = -M_2^{t} y_2 - M_2^t M_1^t y_1 = x_2 + M_2^t x_1, \\ 
            r_{12} & = \langle y_{12}, y_{12}\rangle = \langle y_1,y_1\rangle +
        \langle{M_1 y_2, M_1 y_2}\rangle + 2 \langle M_1 y_2, y_1\rangle \\
            