The double shuffle Lie algebra injects into the Kashiwara-Vergne Lie algebra

In this article we prove that there exists an injective Lie morphism from the double shuffle Lie algebra $\mathfrak{d}\mathfrak{s}$ into the Kashiwara-Vergne Lie algebra $\mathfrak{k}\mathfrak{r}\mathfrak{v}$, forming a commutative triangle with the known Lie injections of the Grothendieck-Teichmüller Lie algebra $\mathfrak{g}\mathfrak{r}\mathfrak{t}\to \mathfrak{d}\mathfrak{s}$ and $\mathfrak{g}\mathfrak{r}\mathfrak{t}\to \mathfrak{k}\mathfrak{r}\mathfrak{v}$.