Poisson brackets and angular momentum I'm trying to find $[M_i, M_j]$ Poisson brackets.
$$\{M_i, M_j\}=\sum_l \left(\frac{\partial M_i}{\partial q_l}\frac{\partial M_j}{\partial p_l}-\frac{\partial M_i}{\partial p_l}\frac{\partial M_j}{\partial q_l}\right)$$
I know that:
$$M_i=\epsilon _{ijk} q_j p_k$$
$$M_j=\epsilon _{jnm} q_n p_m$$
and so:
$$[M_i, M_j]=\sum_l \left(\frac{\partial \epsilon _{ijk} q_j p_k}{\partial q_l}\frac{\partial \epsilon _{jnm} q_n p_m}{\partial p_l}-\frac{\partial \epsilon _{ijk} q_j p_k}{\partial p_l}\frac{\partial \epsilon _{jnm} q_n p_m}{\partial q_l}\right)$$
$$= \sum_l \epsilon _{ijk} p_k \delta_{jl} \cdot \epsilon_{jnm} q_n \delta_{ml}- \sum_l \epsilon_{ijk}q_j \delta_{kl} \cdot \epsilon_{jnm} p_m \delta_{nl}$$
Then I have thought that values that nullify deltas don't add any informations in the summations. And so, $m=l, j=l$ but so I obtain $m=j$. But if $m=l$, the second Levi-Civita symbol in the first summation is zero... And if I go on, I obtain $\{M_i, M_j\}=-p_iq_j$ instead of  $\{M_i, M_j\}=q_ip_j-p_iq_j$
Where am I wrong? Could you  give me some hints to continue? 
 A: You are confusing in the index, such calculations must be carried out very carefully.
I would start with your difention.
$$M_i=\epsilon _{ijk} q_j p_k$$
$$M_p=\epsilon _{pnm} q_n p_m$$
$$\{M_i, M_p\}=\sum_l \left(\frac{\partial M_i}{\partial q_l}\frac{\partial M_p}{\partial p_l}-\frac{\partial M_i}{\partial p_l}\frac{\partial M_p}{\partial q_l}\right)$$
First term 
$=\epsilon _{ijk}p_k\delta_{jl}\epsilon _{pnm}q_n\delta_{ml}=\epsilon _{ilk}p_k\epsilon _{pnl}q_n=(-1)\epsilon _{lik}p_k(-1)^2\epsilon _{lpn}q_n=-\epsilon _{lik}p_k\epsilon _{lpn}q_n=-\left(\delta_{ip}\delta_{kn}-\delta_{in}\delta_{kp}\right)p_kq_n$
Here I used the antisymmetry of $\epsilon _{lik}$ and equation
$\epsilon_{ijk}\epsilon_{imn} = \delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km}$
Second term
Absolutely the same calculations. 
$=\epsilon _{ijk}q_j\delta_{kl}\epsilon _{pnm}p_m\delta_{nl}=\epsilon _{ijl}q_j\epsilon _{plm}p_m=\epsilon _{plm}p_m\epsilon _{ijl}q_j=-\epsilon _{lpm}p_m\epsilon _{lij}q_j=-\left(\delta_{pi}\delta_{mj}-\delta_{pj}\delta_{mi}\right)p_mq_j=$
Make the change $m=k,j=n$. Then
$=-\left(\delta_{pi}\delta_{kn}-\delta_{pn}\delta_{ki}\right)p_kq_n$
All together
$\{M_i, M_p\}=-\left(\delta_{ip}\delta_{kn}-\delta_{in}\delta_{kp}\right)p_kq_n+\left(\delta_{pi}\delta_{kn}-\delta_{pn}\delta_{ki}\right)p_kq_n=\delta_{in}\delta_{kp}p_kq_n-\delta_{pn}\delta_{ki}p_kq_n=p_pq_i-p_iq_p=q_ip_p-p_iq_p$
