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I don't understand the expression (2.2.15). The complete relation would be

$$ \pi_i \pi_j = \frac{1}{2}\left(\left[\pi_i, \pi_j\right] + \left\{\pi_i, \pi_j\right\}\right), $$

where

$$ \vec{\pi} = \vec{p} - \frac{q}{c}\vec{A} $$

in gaussian unit. How can I prove that {πi, πj} = 0?