The discotic nematic (N-D)-biaxial nematic (N-B)-calamitic nematic (N-C) phase transitions are studied in a potassium laurate-decanol-D2O mixture by optical conoscopy. The N-D(+) (N-C(-)) phase is uniaxial positive (negative) as expected. Nevertheless, inside the biaxial range, beyond the known biaxial positive (N-B(+)) nematic phase we have also found a biaxial negative (N-B(-)) nematic phase optically characterized through conoscopic images. A transition point from a N-B(+) to a N-B(-) configuration is determined and discussed in this work. (C) 2013 Elsevier B.V. All rights reserved.