We present a complete description of all the extreme points of the unit ball of $L_s{(}^2{𝑙}_{\mathrm{\infty }}^3)$ which leads to a complete formula for $\parallel f\parallel$ for every $𝑓\in L_s{(}^2{𝑙}_{\mathrm{\infty }}^3){*}^{}$. We also show that ${\mathit{extB}}_{L_s{(}^2{𝑙}_{\mathrm{\infty }}^3)}\subset {\mathit{extB}}_{L_s{(}^2{𝑙}_{\mathrm{\infty }}^{\mathrm{n}})}$ for every n ≥ 4. Using the formula for $\parallel f\parallel$ for every $f\in L_s{(}^2{𝑙}_{\mathrm{\infty }}^3){*}^{}$, we show that every extreme point of the unit ball of $L_s{(}^2{𝑙}_{\mathrm{\infty }}^3)$ is exposed. We also characterize all the smooth points of the unit ball of $L_s{(}^2{𝑙}_{\mathrm{\infty }}^3)$.

Keywords: symmetric bilinear forms on ℝ,³ with the supremum norm, extreme points, exposed points, smooth points