Real vs Complex Representations of a Lie algebra

Proposition 3.39 of Hall's Lie Groups, Lie Algebras And Representations:

"Let $\mathfrak{g}$ be a real Lie algebra, $\mathfrak{g}_\mathbb{C}$ its complexification, and $\mathfrak{h}$ an
arbitrary complex Lie algebra. Then every real Lie algebra homomorphism of $\mathfrak{g}$ into $\mathfrak{h}$ extends uniquely to a complex Lie algebra homomorphism of $\mathfrak{g}_\mathbb{C}$ into $\mathfrak{h}$."

In particular this means that any real representation of $\mathfrak{g}$ defines a complex representation of $\mathfrak{g}_\mathbb{C}$.

Question: Does the converse hold? Does any complex representation of $\mathfrak{g}_\mathbb{C}$ define a real representation of $\mathfrak{g}$? Are there any conditions for when this may or may not hold?

Answer

Of course it does. If the Lie algebra $\mathfrak{g}_{\mathbb C}$ acts on a complex vector space $V$, simply consider the restriction of this action to $\mathfrak g$. To be more precise, if $X\in\mathfrak g$ and if $v\in V$, then define $X.v$ as $(X\otimes 1).v$. I am assuming here that Hall defined $\mathfrak{g}_{\mathbb C}$ as $\mathfrak{g}\bigotimes_{\mathbb{R}}\mathbb{C}$. If he used another definition, pleas say which one does he use.