Clarification for the terminology "complex representation" of "real" Lie algebra I am reading the book "Lie Groups, Lie Algebras and Representations: An elementary introduction" by B. C. Hall. There, the author defines what is meant by (finite-dimensional) representations of Lie groups and Lie algebras.
Particularly, he states:
Definition: If $\mathfrak{g}$ is a real or complex Lie algebra, then a finite-dimensional complex representation of $\mathfrak{g}$ is a a Lie algebra homomorphism $\pi: \mathfrak{g} \rightarrow \mathfrak{gl} \left( V \right)$, where $V$ is a finite dimensional complex vector space.
Here, $\mathfrak{gl} \left( V \right)$, is identified with the Lie algebra of matrices (by choosing a basis on $V$).
Now, the definition is clear when $\mathfrak{g}$ is a complex Lie algebra. However, if $\mathfrak{g}$ is a real Lie algebra, how does this definition work? For $\pi: \mathfrak{g} \rightarrow \mathfrak{gl} \left( V \right)$ to be a Lie algebra homomorphism, it is required that $\pi$ be a linear transformation first. But since $V$ is complex, $\mathfrak{gl} \left( V \right)$ is complex and therefore, $\pi$ cannot be a linear transformation (since the domain and the codomain are defined over different fields).
To make sense of this definition, I have two things in my mind:
(1) We consider the vector space $V$ to be real but with double dimension. However, this does not really suit the needs. For then, it coincides with the definition of a real representation.
(2) We only require $\pi$ to be real-linear, instead of a (truly) linear transformation. This would make more sense since then there would be a difference between real representations and complex representations of the same real Lie algebra.
Am I on the right path to understand the concept?
 A: The point here is what you mean by $\mathfrak{gl}(V)$. You don't really have to work with matrices here, since you can view $\mathfrak{gl}(V)$ as the space of linear maps from $V$ to itself with the commutator as a bracket, i.e. $[f,g]=f\circ g-g\circ f$. When talking about complex representations, you always require that $V$ is a complex vector space and that $\mathfrak{gl}(V)$ is interpreted as the space $\mathfrak{gl}_{\mathbb C}(V)$ of complex linear maps from $V$ to itself.
Now $\mathfrak{gl}_{\mathbb C}(V)$ is a complex Lie algebra and thus also a real Lie algebra. The standard convention then is that by a complex representation of a real Lie algebra $\mathfrak g$ one means a homomorphism $\mathfrak g\to \mathfrak{gl}_{\mathbb C}(V)$ of real Lie algebras. For a complex Lie algebra $\mathfrak g$, one defines a complex representation as a a homomorphism $\mathfrak g\to \mathfrak{gl}_{\mathbb C}(V)$ of complex Lie algebras, i.e. one in addition requires that the map $\mathfrak g\to \mathfrak{gl}_{\mathbb C}(V)$ is linear over the complex numbers. (So there is a difference between complex representations of a complex Lie algebra and complex representations of the underlying real Lie algebra.)
This can also be nicely expressed in the picutre of a representation via an action $\mathfrak g\times V\to V$. If $\mathfrak g$ is real, one requires that the action map is complex linear in the second varible, whereas if $\mathfrak g$ is complex, one requires the map to be complex bilinear.
