  The derivation of the self-dual relations for the two-form gauge field in the
Nambu-bracket description of M5-brane in a constant C-field background
initiated in arXiv:0907.4596 is completed by including contributions from all
the fields in the M5-brane action. The result is used to examine Seiberg-Witten
map of the BPS conditions for the string solitons, up to the first order in the
expansion by the parameter $g$ which characterizes the strength of the
interactions through the Nambu-bracket.
