Notes on Kashiwara-Vergne and double shuffle Lie algebras

We explain the current situation of the relationship between the Kashiwara-Vergne Lie algebra $\mathfrak{k}\mathfrak{r}\mathfrak{v}$ and the double shuffle Lie algebra $\mathfrak{d}\mathfrak{m}\mathfrak{r}$. We also show the validity of Ecalle’s senary relation for small depths.