I want to check if the fundamental representation of $O(3)$ is irreducible on $\mathbb{R}^3$ and $\mathbb{C}^3$. I want to use isomorphism properties.
I know this relation exists $$ \mathfrak{o}(3)\sim \mathfrak{so}(3) \sim \mathfrak{su}(2) $$
and I know that the irreps of $\mathfrak{su}(2)$ comes from irreps of $\mathfrak{gl}(2,\mathbb{C})$.
So, I would say that $\mathfrak{o}(3)$ has 3-dimentional irrep on $\mathbb{R}^3$, the same of $\mathfrak{su}(2)$, but I'm not sure.
How are the representation related by algebra isomorphism?
Let be $D$ an irreducible representation of $\mathfrak{su}(2)$ on $\mathbb{C}^3$.
How can I construct the equivalent representation $D'$ of $\mathfrak{o}(3)$ starting from $D$? Is D' a representation on $\mathbb{R}^3$ or $\mathbb{C}^3$?