On page 12 of [the paper](http://download.springer.com/static/pdf/137/art%253A10.1007%252FBF02941339.pdf?originUrl=http%3A%2F%2Flink.springer.com%2Farticle%2F10.1007%2FBF02941339&token2=exp=1486462161~acl=%2Fstatic%2Fpdf%2F137%2Fart%25253A10.1007%25252FBF02941339.pdf%3ForiginUrl%3Dhttp%253A%252F%252Flink.springer.com%252Farticle%252F10.1007%252FBF02941339*~hmac=7569d8e384d349cb8381c29d2fcd4bcc3d91ee9e45ad56c32446e6ed463e62dc), there is a formula about super Poisson bracket on a Lie super group $G$:
\begin{align}
\{\phi, \psi\} = \sum\_{\mu, \nu} (-1)^{|\phi||\nu|} r^{\mu \nu} ( R\_{\mu} \phi R\_{\nu} \psi - L\_{\mu} \phi L\_{\nu} \psi ),
\end{align}
where $\phi, \psi$ are functions on $G$.


I am trying to verify that the bracket satisfies
\begin{align}
\{\psi, \phi\} = - (-1)^{|\psi| |\phi|} \{\phi, \psi\}.
\end{align}


We have 
\begin{align}
\{\psi, \phi\} & = \sum\_{\mu, \nu} (-1)^{|\psi||\nu|} r^{\mu \nu} ( R\_{\mu} \psi R\_{\nu} \phi - L\_{\mu} \psi L\_{\nu} \phi ) \\
& = \sum\_{\mu, \nu} (-1)^{|\psi||\mu|} r^{\nu \mu} ( R\_{\nu} \psi R\_{\mu} \phi - L\_{\nu} \psi L\_{\mu} \phi ) \\
& = \sum\_{\mu, \nu} (-1)^{|\psi||\mu|} r^{\nu \mu} ( -(-1)^{|R\_{\nu}\psi | | R\_{\mu} \phi|} R\_{\mu} \phi R\_{\nu} \psi + (-1)^{|L\_{\nu} \psi| |L\_{\mu} \phi|} L\_{\mu} \phi L\_{\nu} \psi ) \\
& = \sum\_{\mu, \nu} (-1)^{|\psi||\mu|} (-1)^{|\mu||\nu|} r^{ \mu \nu} ( (-1)^{|R\_{\nu}\psi | | R\_{\mu} \phi|} R\_{\mu} \phi R\_{\nu} \psi - (-1)^{|L\_{\nu} \psi| |L\_{\mu} \phi|} L\_{\mu} \phi L\_{\nu} \psi ).
\end{align}
Therefore
\begin{align}
& (-1)^{|\psi||\phi|}\{\psi, \phi\} \\
& = \sum\_{\mu, \nu} (-1)^{|\psi||\phi|} (-1)^{|\psi||\mu|} (-1)^{|\mu||\nu|} r^{ \mu \nu} ( (-1)^{|R\_{\nu }\psi | | R\_{\mu} \phi|} R\_{\mu} \phi R\_{\nu} \psi - (-1)^{|L\_{\nu} \psi| |L\_{\mu} \phi|} L\_{\mu} \phi L\_{\nu} \psi ).
\end{align}
Therefore we need to show that
\begin{align}
& (-1)^{|\psi||\phi| + |\psi||\mu| + |\mu||\nu| + |R\_{\nu }\psi | | R\_{\mu} \phi| + |\phi| |\nu| } = 1, \\
& (-1)^{|\psi||\phi| + |\psi||\mu| + |\mu||\nu| + |L\_{\nu }\psi | | L\_{\mu} \phi| + |\phi| |\nu| } = 1.
\end{align}
In general, do we have the above identities? How to prove them? Thank you very much.



How to prove a bracket is super anti-commutative?
This question was solved by Vladimir Dotsenko in the comments of the [question](https://mathoverflow.net/questions/261506/construct-super-poisson-brackets-on-the-coordinate-rings-of-lie-super-groups).


