# Two definitions of the super Jacobi identity

In this paper, page 149, the super Jacobi identity is given by \begin{align} J(x, y,z) := (-1)^{|x||z|}[[x, y],z] +(-1)^{|z||y|}[[z,x], y]+(-1)^{|y||x|}[[y,z],x] = 0. \end{align} But in this article, the super Jacobi identity is given by \begin{align} [x,[y,z]]=[[x,y],z]+(-1)^{|x| |y|}[y,[x,z]]. \end{align} Are these two definitions equivalent? Thank you very much.

By super skew-symmetry $[x, y] = - (-1)^{|x||y|}[y, x]$. Also note that $\left|[x, y]\right| = |x| + |y| \bmod 2$, in particular, $(-1)^{|[x, y]|} = (-1)^{|x|+|y|}$. So we see that
I believe these should be the same. The reason being, $J(x,y,z)$ reduces to the ordinary Jacobi identity in all cases except one: when any two of the elements in $\{x,y,z\}$ are Fermi and the third one is Bose, in which case one of the three (usual) Jacobi terms has its sign flipped.