Summary: Recently, there was a distance function d studied in [the authors et al., Linear Nonlinear Anal. 5, No. 3, 421–437 (2019; Zbl 1454.90056)], which was shown a semi-distance in the setting of symmetric cone. This result was indeed verified under the assumption of operator commute. it was further proved that it could become a proximal distance in the setting of second-order cone. In this short paper we improve these results by showing that the distance function d studied in [loc. cit.] is not only a semi-distance, but also a proximal distance in the setting of symmetric cone without assuming operator commute.