Eulerian idempotent and Kashiwara-Vergne conjecture
  By using the interplay of the Eulerian idempotent and the Dynkin idempotent, we construct explicitly a particular symmetric solution (F,G) of the first equation of the Kashiwara-Vergne conjecture: x+y-log(exp(y)exp(x))=(1-exp(-ad x))F(x,y)+(exp(ad y)-1)G(x,y) . Then, we explicit all the solutions of the equation in the completion of the free Lie algebra generated by two indeterminates x and y thanks to the kernel of the Dynkin idempotent. 
Emily Burgunder