ch spaces with their norm induced metrics. Although this would be a consequence of the mentioned result from [@godefroy2003lipschitz] which states that a Banach space has $\lambda$-BAP if and only if ${\mathcal{F}}(X)$ does, actually the order of the proofs is reversed. First, Godefroy and Kalton prove that, for finite dimensional Banach spaces $X$, ${\mathcal{F}}(X)$ has the MAP, and then use this to prove the result for general Banach spaces. The proof of the finite dimensional part also involves harmonic averaging.

Since Lipschitz-free spaces over finite dimensional Banach spaces even possess a Schauder basis ([@HaPe]), we suggest to turn the attention to locally compact metric groups that are very closely related to such Banach spaces; that is, Carnot groups. We note that a Carnot group is both analytically and algebraically a mild generalization of a finite dimensional Banach space and many results from geometric analysis on Euclidean spaces have been generalized to the setting of Carnot groups (see [@LeD]). The duals of Lipschitz-free spaces over Carnot groups have been already investigated in [@candido2019isomorphisms] and their Lipschitz-free spaces in [@albiac2020lipschitz].

Let $G$ be a Carnot group with a Carnot-Carath' eodory metric. Does ${\mathcal{F}}(G)$ has the Schauder basis?

In the case of connected Lie groups, one has a canonical compatible left-invariant metric: the left-invariant Riemannian metric. Since such a metric is locally bi-Lipschitz equivalent to the Euclidean metric, and isomorphic properties of Lipschitz-free spaces often depend only on the local behaviour of the metric, the following question is very natural. We note that one could even replace Lie group there with a general connected Riemannian manifold (without boundary).

\[quest:liegrp\] Let $G$ be a connected (real) Lie group equipped with a left-invariant Riemannian metric. Do we have ${\mathcal{F}}(G)\simeq {\mathcal{F}}({\mathbb{R}}^n)$, where $n$ is the dimension of the (real) Lie algebra $\mathfrak{g}$ of $G$?

Still 