This is perhaps a stupid question. We consider $G =\text{SU}(3)$ and $\pi : G \to \textrm{GL}(\mathfrak{g})$ the adjoint representation that sends $g \in G$ to $Ad_g$ that acts on the Lie algebra $\mathfrak{g}$ be the following formula. For $X \in \mathfrak{g}$, $$Ad_g(X) = gXg^{-1}.$$
Now I am trying to find the weight vectors for this representation. I get that they are the elementary matrices $E_{ij}$ but there is one problem: These $E_{ij}$ technically don't live in $\mathfrak{g}$! This is because an elementary matrix $E_{ij}$ does not satisfy the relation $X + X^\ast = 0$. What am I misunderstanding here?
Yes, absolutely, the issue is that eigenvectors (weight vectors) often lie only in the complexified Lie algebra. Here, a useful, compatible model of the complexification of $\frak su$$(n)$ is $\frak sl$$(n,\mathbb C)$, making the relevance of @TobiasKildetoft's comments perhaps clearer. Indeed, the question does turn into asking about the action of the complexified algebra on itself.