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An irreducible representation $\phi:\mathfrak{g}_\mathbb{C} \rightarrow GL(\mathbb{C,n})$ is a Lie algebra homomorphism by definition. That is, it's linear. So, to consider your example, we see that for $x,y \in \mathfrak{so}(3)$, \begin{align} \pi(x+iy) = \pi(x)+i\pi(y) \end{align} which shows that working with \begin{align} \pi: \mathfrak{so}(3) \...



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