Why $\mathfrak{g} \subseteq V \otimes V^\ast$?

Question

I'm reading Chris Schommer-Priesthis's answer to What is torsion in differential geometry intuitively?.

I know that $\mathfrak{g}$ is the tangent space of G at $e$ but don't understand why $\mathfrak{g} \subseteq V \otimes V^\ast$.

Answer 1

It is perhaps more useful to think of $V\otimes V^{*}$ linear maps from $V$ to $V$, with the commutator given by $[f,g]=f\circ g-g\circ f$. This is (isomorphic to) the Lie algebra of $GL(V)$.