Let d_*(1, w)² = ℝ² with the octagonal norm of weight w. It is the two dimensional real predual of Lorentz sequence space. In this paper we classify the smooth points of the unit ball of the space of symmetric bilinear forms on d_*(1, w)². We also show that the unit sphere of the space of symmetric bilinear forms on d_*(1, w)² is the disjoint union of the sets of smooth points, extreme points and the set A as follows: SLs(d2*(1,w)2)=smBLs(d2*(1,w)2)∪extBLs(d2*(1,w)2)∪A,

where the set A consists of ax₁x₂ + by₁y₂ + c(x₁y₂ + x₂y₁) with (a = b = 0, c=±11+w2), (a ≠ b, ab ≥ 0, c = 0), (a = b, 0 < ac, 0 < |c| < |a|), (a ≠ |c|, a = −b, 0 < ac, 0 < |c|), (a=1-w1+w,b=0,c=11+w), (a=1+w+w(w2-3)c1+w2,b=w-1+(1-3w2)cw(1+w2),12+2w<c<1(1+w)2(1-w),c≠11+2w-w2), (a=1+w(1+w)c1+w,b=-1+(1+w)cw(1+w),0<c<12+2w) or ( a=1-w(1+w)c1+w,b=1-(1+w)c1+w,11+w<c<1(1+ω)2(1-ω)).

Keywords:

Symmetric bilinear forms, extreme points, ,smooth points, the octagonal norm on $mathbb{R}^2$ with weight ,$w$, the two dimensional ,real predual of the Lorentz sequence space.