In this book I read
Proposition A.1. The irreducible complex representations of a real Lie algebra $\mathfrak{g}$ are in one-to-one correspondence with the irreducible complex-linear representations of its complexification $\mathfrak{g}_C$
How can this be correct?
As an example take $\mathfrak{so}(3)$, which is a real Lie algebra. Naturally, the $3$-dimensional representation of $\mathfrak{so}(3)$ is a real representation
$$ \pi : \mathfrak{so}(3) \rightarrow Gl(R^3) $$
We can complexify the representation by considering
$$ \pi : \mathfrak{so}(3) \rightarrow Gl(C^3) $$
This means we have the same $3\times 3$ matrices, but now they act on complex $3$-dimensional vectors. (This is Example 5.32 at page 249 in the book I linked to above. The author writes there: "the complexification of the fundamental representation of $\mathfrak{so}(3)$ is just given by the usual $\mathfrak{so}(3)$ matrices acting on $C^3$ rather than $R^3$.
Alternatively, we can consider the complexified Lie algebra $\mathfrak{so}(3)_C$. This means we now allow complex linear combination of the $\mathfrak{so}(3)$ elements:
$$ \mathfrak{so}(3)_C= \{ x + iy | x,y \in \mathfrak{so}(3) \}$$
The representations of $\mathfrak{so}(3)_C$ are maps to complex vector spaces, for example
$$ \pi : \mathfrak{so}(3)_C \rightarrow Gl(C^3) $$
These are the complex linear combinations of the usual $\mathfrak{so}(3)$ matrices acting on complex vectors.
How can this representation be in "in one-to-one correspondence" to $ \pi : \mathfrak{so}(3) \rightarrow gl(C^3) $?