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.

According to

[2], the original paper is

[3]. In this paper there is some detail but it is mostly stated that the calculation is tedious and it refers to another
paper which I cannot access. In a recent paper

[4] there is more information, but

[4] is still very long.
Also

[4] uses a regulator and I think that in principle, depending on its exact use,
the use of the regulator could lead to incorrect results.
There is more detail in lecture notes by 't Hooft

[5], but the bulk of the calculation is still left as an exercise.
Anyway, I think Zwiebach is wise to not include the calculation in his book. Even if I finished the calculation of \eqref{eq:20170103}, I would not learn much,
so I decided to stop trying to prove \eqref{eq:20170103}.

In the calculation of \eqref{eq:20170103} many terms will cancel and the right hand side of \eqref{eq:20170103} are some terms that survive the cancellations.
What I would like to have is some way to calculate \eqref{eq:20170103} in which those cancellations do not ''miraculously'' happen. In general, I think that if one does a calculation and many terms cancel, so that
the final result is surprisingly simple, that often means one does not have sufficient understanding about what is going on.

A similar situation occurs in checking that
\begin{equation*}
[p^{-} , p^I ] = [ p^{-} , p^{+} ] = 0
\end{equation*}
In this case the calculation is short. The right hand side is indeed zero, so the quantum string theory is translation invariant. Although in this case the
calculation is very short, I am not satisfied that there is no better way to understand this calculation in the light-cone gauge.

References
[1]
A First Course in String Theory, Zwiebach, 2009

[2]
A footnote in Lectures on String Theory by David Tong

[3]
Goddard, Goldstone Rebbi and Thorn “Quantum Dynamics of a Massless Relativistic String”, Nucl. Phys. B56, (1973)

[4]
A Note on Angular Momentum Commutators in Light-Cone Formulation of Open Bosonic String Theory, Bering,
arXiv:1104.4446v2

[5] Introduction to String Theory, 't Hooft

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