I am studying Hall's book on "Lie Groups, Lie Algebras and Representations", and I came across this definition:
Definition: A real Lie algebra $\mathfrak{g}$ is said to admit a complex structure if there is a "multiplication by $\iota$" map $J: \mathfrak{g} \rightarrow \mathfrak{g}$ that makes $\mathfrak{g}$ into a complex vector space in such a way that the bracket $\left[ \cdot, \cdot \right] : \mathfrak{g} \times \mathfrak{g} \rightarrow \mathfrak{g}$ is complex bilinear.
What I do not understand is how does a map ($J$ in this case) make $\mathfrak{g}$ into a complex vector space? Or, is it the case that the author means to say the set $\left\lbrace X + \iota JY: X, Y \in \mathfrak{g} \right\rbrace$ is a complex vector space?
