In [this paper](http://link.springer.com/article/10.1007/BF02941339), 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](https://en.wikipedia.org/wiki/Lie_superalgebra), 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.



Two definitions of the super Jacobi identity
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


\begin{align\*}
[x, [y, z]] &= -(-1)^{|x||[y, z]|}[[y, z], x]\\
&= -(-1)^{|x||y| + |x||z|}[[y, z], x]\\
&= -(-1)^{|x||z|}\left(-(-1)^{|x||z|}[[x, y],z] -(-1)^{|z||y|}[[z,x], y]\right)\\
&= [[x, y], z] + (-1)^{|x||z| + |y||z|}[[z, x], y]\\\
&= [[x, y], z] - (-1)^{|x||z| + |y||z|}(-1)^{|[z, x]||y|}[y, [z, x]]\\
&= [[x, y], z] - (-1)^{|x||z| + |y||z|}(-1)^{|x||y| + |y||z|}[y, [z, x]]\\
&= [[x, y], z] - (-1)^{|x||y| + |x||z|}[y, [z, x]]\\
&= [[x, y], z] - (-1)^{|x||y| + |x||z|}[y, -(-1)^{|x||z|}[x, z]]\\
&= [[x, y], z] + (-1)^{|x||y|}[y, [x, z]].
\end{align\*} 


