I am wondering if anyone could explain to me either why my method is not valid or point out where I have made an algebraic slip. I have been looking at this for a long time, to no avail.
Let $\{\cdot , \cdot\}$ be a Poisson bracket.
I wish to show that for angular momentum $M=x\times p$, $\{M_i,M_j\}=\varepsilon_{ijk} M_k$
I have previously shown that $\{x_i,M_j\}=\varepsilon_{jik} x_k$ and $\{p_i,M_j\}=\varepsilon_{jik} p_k$
So I proceed as follows:
$\{M_i,M_j\}=\varepsilon_{iab} \{x_a p_b,M_j\}$ $= \varepsilon_{iab} (x_a\{p_b,M_j\}+\{x_a,M_j\}p_b)$ $= \varepsilon_{iab} (x_a\varepsilon_{jbk} p_k+\varepsilon_{jak} x_kp_b)$ $=(\delta_{ik}\delta_{aj}-\delta_{ij}\delta_{ak})x_a p_k+ (\delta_{bk}\delta_{ij}-\delta_{bj}\delta_{ik})x_k p_b$ $=x_jp_i-x_kp_k+x_bp_b-x_ip_j$ $=x_jp_i-x_ip_j$ $=\varepsilon_{kji}M_k$
which is minus the result I should be getting.
(I know that I could have expanded both $M_i,M_j$ and prove it that way, but I don't see any reason why this method should not work.)
Thank you.