  We discuss the importance of $Z_2$ symmetry of large $N=4$ superconformal
algebra for finding its free field realization that is more complete than the
most commonly used one, that is, the Sevrin, Troost and van Proeyen
realization. This in turn has implications for the search of a conformal field
theory dual to superstrings moving in $AdS_3 \times S^3 \times S^3 \times S^1$
background.
