Obtaining representation of a real Lie algebra from the complexification "by restriction"

Let $\mathfrak{g}$ be a Lie algebra, then $\mathfrak{g}\otimes_{\mathbb{R}}\mathbb{C}$ is a complex Lie algebra where we extend the Lie bracket by complex linearity. The map $i : \mathfrak{g} \to \mathfrak{g}\otimes_{\mathbb{R}}\mathbb{C}$ given by $X \mapsto X\otimes 1$ is a Lie algebra homomorphism.

Recall that a representation of a Lie algebra is just a Lie algebra homomorphism to $\mathfrak{gl}(V)$. Therefore, if $\rho : \mathfrak{g}\otimes_{\mathbb{R}}\mathbb{C} \to \mathfrak{gl}(V)$ is a representation of $\mathfrak{g}\otimes_{\mathbb{R}}\mathbb{C}$, then $\rho\circ i : \mathfrak{g} \to \mathfrak{gl}(V)$ is a representation of $\mathfrak{g}$ (because the composition of Lie algebra homomorphisms is a Lie algebra homomorphism).