Let $G$ be a Carnot group (aka stratified group), so that $G$ is a connected and simply connected finite-dimensonal Lie group, whose Lie algebra $\mathfrak{g}$ admits a decomposition $\mathfrak{g} = V_1 \oplus \dots \oplus V_n$ where $[V_1, V_k] = V_{k+1}$. Let $\mathcal{H} \subset TG$ be the left-invariant "horizontal distribution" which agrees with $V_1$ at the identity. Then $\mathcal{H}$ is bracket generating. If we fix an inner product $\langle \cdot, \cdot \rangle$ on $V_1$, we obtain a left-invariant sub-Riemannian metric $g$ on $G$.

Let $d$ be the corresponding Carnot–Carathéodory distance on $G$. That is, the distance $d(x,y)$ is the length of the shortest horizontal path joining $x$ to $y$.

If $G = \mathbb{H}^{2m+1}$ is a Heisenberg group, i.e. $n=2$ and $\dim V_2 = 1$, then an "explicit" formula for $d$ is known [1], for any inner product on $V_1$. (As Richard Montgomery points out, it is not explicit in the strongest sense because it is in terms of a solution of a transcendental equation. But that sort of thing is explicit enough for my purposes.)

There are also "explicit" formulas known in case $(G, \langle \cdot,\cdot \rangle)$ is H-type in the sense of Kaplan [2]; see [3,4,5].

Are there any other (classes of) Carnot groups for which we know how to "explicitly" compute the distance $d$?

[1]. *Beals, Richard W.; Gaveau, Bernard; Greiner, Peter C.*, **Hamilton-Jacobi theory and the heat kernel on Heisenberg groups**, J. Math. Pures Appl., IX. Sér. 79, No.7, 633-689 (2000). ZBL0959.35035.

[2]. *Kaplan, Aroldo*, **Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms**, Trans. Am. Math. Soc. 258, 147-153 (1980). ZBL0393.35015.

[3].
*Korányi, Adam*, **Geometric properties of Heisenberg-type groups**, Adv. Math. 56, 28-38 (1985). ZBL0589.53053.

[4] *Rigot, Séverine*, **Counter example to the Besicovitch covering property for some Carnot groups equipped with their Carnot-Carathéodory metric**, Math. Z. 248, No. 4, 827-848 (2004). ZBL1082.53030.

[5]. *Tan, Kang-Hai; Yang, Xiao-Ping*, **Characterisation of the sub-Riemannian isometry groups of $H$-type groups**, Bull. Aust. Math. Soc. 70, No. 1, 87-100 (2004). ZBL1070.53013.