Classification of the extreme points of (Formula presented) by computation

Let (Formula presented) be endowed with the supremum norm. In [Comment. Math. 2017, 57 (1), 1–7], S.G. Kim classified the extreme points of the unit ball of (Formula presented) only using Mathematica 8, where (Formula presented) is the space of symmetric bilinear forms on (Formula presented). It seems to be interesting and meaningful to classify the extreme points of the unit ball of (Formula presented) without using Mathematica 8. The aim of this paper is to make such classification by mathematical calculations.