We recalculate, in a systematic and pedagogical way, one of the most important results of Bosonic open string theory in the light-cone formulation, namely the [J-i,J-j] commutators, which together with Lorentz covariance, famously yield the critical dimension D = 26 and the normal order constant a = 1. We use traditional transverse oscillator mode expansions (avoiding the elegant but more advanced language of operator product expansions). We streamline the proof by introducing a novel bookkeeping/regularization parameter kappa to avoid splitting into creation and annihilation parts, and to avoid sandwiching between bras and kets.
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