Why is the Lie algebra corresponding to group SO(3) called o(3)?

Question

How is Lie algebra named? Why is it $\mathfrak{su}(2)$ for group $SU(2)$, but $\mathfrak{o}(3)$ for group $SO(3)$? What does the "$\mathfrak{s}$" in algebra $\mathfrak{su}(2)$ mean?

Answer 1

The "s" stands for special. Lie algebras are usually named after Lie groups. $\text{U}(2)$ and $\text{SU}(2)$ have different Lie algebras so they're named $\mathfrak{u}(2)$ and $\mathfrak{su}(2)$, but as it turns out, $\text{O}(3)$ and $\text{SO}(3)$ have the same Lie algebra, so $\mathfrak{o}(3) \cong \mathfrak{so}(3)$.