Products of conjugacy classes in ${\text{SL}}_2(k)$ and ${\text{PSL}}_2(k)$

Let $k$ be a field with $u$-invariant $\le 2$. Assume further that $k$ is not quadratically closed, $\mathsf{c}\mathsf{h}\mathsf{a}\mathsf{r}(k)\mathrm{\ne }2$ and $\mathrm{\mid }k\mathrm{\mid }\ge 5$. It is known that the covering number of both ${\text{SL}}_2(k)$ and ${\text{PSL}}_2(k)$ is three, while their extended covering number is four. In this article, we completely describe the product of two conjugacy classes in ${\text{SL}}_2(k)$ and ${\text{PSL}}_2(k)$. Further, we also describe the product of three conjugacy classes (at least two of which are distinct) in ${\text{SL}}_2(k)$ and ${\text{PSL}}_2(k)$.