EditSecond Edit:
Just out of curiosity: For $(a,b,c)=(1,2,2k+1)$, so $c \ge 3$ is odd, we get using the sum of angles in a triangle:
$$\operatorname{acos}(\frac{4 \, c^{5} + 28 \, c^{4} + 62 \, c^{3} + 2 \, c^{2} - 153 \, c - 135}{12 \, {\left(c + 2\right)}^{3} {\left(c + 1\right)} c} ) +$$
$$ \operatorname{acos}(\frac{14 \, c^{5} + 98 \, c^{4} + 226 \, c^{3} + 142 \, c^{2} - 135 \, c - 153}{18 \, {\left(c^{2} + 2 \, c - 1\right)} {\left(c + 2\right)}^{2} {\left(c + 1\right)}}) + $$
$$\operatorname{acos}(\frac{4 \, c^{6} + 24 \, c^{5} + 70 \, c^{4} + 156 \, c^{3} + 187 \, c^{2} - 18 \, c - 135}{12 \, {\left(c^{2} + 2 \, c - 1\right)} {\left(c + 2\right)} {\left(c + 1\right)}^{2} c}) = \pi$$
Third edit:
I think the main property which distinguishes $d$ for example from the Jaccard or other metrics is the proven property ( https://mathoverflow.net/a/342921/6671) :
For all $a \neq b, a\neq c$ we have:
$$d(a,b)+d(a,c) > 1$$
I have tested other metrics with this property and they also seem to embedd in Euclidean Space. On the other hand metrics who do not have this property do not embedd. So I think this is the point to be taken into consideration.
If someone has an idea how to exploit this property that would be very nice!
