A magnetostatic exercise in 10 dimensions

I calculate the electromagnetic field generated by electrical currents in 10 spacetime dimensions (9 space and 1 time). The set up is as follows: the current flows down the positive $x_1$-axis, hits the origin and then spreads out isotropically in the $x_2 x_3 x_4$ subspace, see figure 1 and 2. I wanted to calculate this because in string theory a similar calculation is needed to obtain the Kalb-Ramond field generated by a string ending on a $D3$-brane [1]

A calculation in magnetostatics

I wanted to calculate the magnetic field generated by a current which flows down the positive $z$-axis, hits the origin and then spreads out radially over the $xy$ plane, see figure 1.

Lorentz invariance of string theory in the light-cone gauge

On page 261 in his book [1] Zwiebach writes ''There is much at stake in this calculation. It is in fact, one of the most important calculations in string theory. [...] The calculation is long and uses many of our previously derived results''. Then Zwiebach states the result \begin{align} \left[ M^{- I}, M^{- J}\right] = &- \frac{1}{\alpha'\ { p^+ }^2} \sum_{m=1}^{\infty} \left(\alpha^I_{-m} \alpha^J_m - \alpha^J_{-m} \alpha^I_m \right)\nonumber\\ &\times \left\{ m \left[1 - \dfrac{1}{24} (D-2) \right] + \dfrac{1}{m} \left[ \dfrac{1}{24} (D-2) + a \right] \right\}\label{eq:20170103} \end{align} This is the commutator of two Lorentz transformations in the light-cone gauge. The commutator should be zero for string theory to be Lorentz invariant. The calculation of the commutator is indeed very tedious and after scribling too many pages I gave up. I googled for a quick way to obtain \eqref{eq:20170103}. Here are the more interesting results that I found.