Double shuffle relations imply the infinitesimal hexagon equation

Double shuffle Lie algebra $\mathfrak{dmr}_0$ was introduced by G.~Racinet in the algebraic study of the multiple zeta values. In this note, we prove that for any $\psi\in \mathfrak{dmr}_0$, it satisfies the infinitesimal hexagon equation $[\psi(x,y),x]+[\psi(-x-y,y),-x-y]=0$. The proof is through the comparison of two different Hopf algebras.