= Solution
In the <symmetric gauge>,
$$
\partial_iA_j+\partial_jA_i=0,
\qquad
\partial_iA_j-\partial_jA_i
=\varepsilon_{ijk}B_k.
$$
Together with $[p_i,f(x)]=-i\hbar\partial_i f$, this gives
$$
\begin{aligned}
[p_i+eA_i,p_j-eA_j]
&=-e[p_i,A_j]+e[A_i,p_j]\\
&=ie\hbar(\partial_iA_j+\partial_jA_i)=0,
\end{aligned}
$$
whereas
$$
\begin{aligned}
[p_i+eA_i,p_j+eA_j]
&=e[p_i,A_j]+e[A_i,p_j]\\
&=ie\hbar(\partial_jA_i-\partial_iA_j)\\
&=-ie\hbar\varepsilon_{ijk}B_k.
\end{aligned}
$$
Thus
$$
\boxed{[p_i+eA_i,p_j-eA_j]=0},
\qquad
\boxed{[p_i+eA_i,p_j+eA_j]
=-ie\hbar\varepsilon_{ijk}B_k}.
$$
These are the commutators of the <kinetic momentum and magnetic pseudomomentum>.
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