Addendum to “Violating the string winding number maximally in anti–de Sitter space”

We revisit the computation of maximally winding violating string amplitudes in three-dimensional anti–de Sitter space discussed in Ref. G. Giribet, Violating the string winding number maximally in anti-de Sitter space, Phys. Rev. D 84, 024045 (2011). Here, we give an alternative derivation of these observables. This derivation, the simplest to the best of our knowledge, follows from identities between spectrally flowed representations of $\widehat{sl}(2{)}_k$ Kac-Moody algebra and, in contrast to G. Giribet, Violating the string winding number maximally in anti-de Sitter space, Phys. Rev. D 84, 024045 (2011), it does not resort to conjugate representations with auxiliary fields, but it rather involves the standard Wakimoto free field representation.