Reduced coaction Lie algebra, double shuffle Lie algebra and noncommutative krv2 equation

We study the reduced coaction Lie algebra $\mathfrak{rc}_0$, which is defined by an algebraic equation satisfied by the reduced coaction (an upgraded version of the necklace cobracket) and the skew-symmetric condition. We prove that the double shuffle Lie algebra $\mathfrak{dmr}_0$ together with the skew-symmetric condition injects to $\mathfrak{rc}_0$, and that $\mathfrak{rc}_0$ together with the krv1 equation injects to the Kashiwara-Vergne Lie algebra $\mathfrak{krv}_2$.