Box-quasimetrics and horizontal joinability on Cartan groups

On a Cartan group $\mathbb K$ equipped with a Carnot–Carathéodory metric $d_{cc}$, we find the exact value of a constant in the $(1,q_2)$-generalized triangle inequality for its Box-quasimetric. It is proved that any two points $x,y\in\Bbb K$ can be joined by a horizontal $k$-broken line $L^k_{x,y}$, $k\leq 6$; moreover, the length of such a broken line $L^k_{x,y}$ does not exceed the quantity $Cd_{cc}(x,y)$ for some constant $C$ not depending on the choice of $x,y\in\mathbb K$. The value $6$ here is nearly optimal.