I have encountered the following definition on page 83 of Structure and Geometry of Lie Groups, by Hilgert & Neeb:
"Let $\mathfrak{g}$ be a Lie algebra, and $V$ be a vector space. Suppose that $\mathfrak{g}\times V\rightarrow V,\;\;\;(x,v)\mapsto x\cdot v$ is a bilinear map. If $[x,y]\cdot v=x\cdot(y\cdot v)-y\cdot(x\cdot v)\;\;\;\text{ for }\;\;\;x,y\in\mathfrak{g},v\in V,$ then $V$ is called a $\mathfrak{g}$-module. "
In the same way that a vector space is defined as a triple $(V,+,\cdot)$, where '$\cdot$' is scalar multiplication, shouldn't a $\mathfrak{g}$-module be defined with respect to the map $(x,v)\mapsto x\cdot v$ as its 'scalar multiplication' map? In other words, shouldn't the definition end with "then $(V,+,\cdot)$ is called a $\mathfrak{g}$-module." Or am I missing something?
Thanks in advance :)