Let $G$ be a Lie group with Lie algebra $\mathfrak {g}.$ Let $\omega$ be a left-invariant non-degenerate closed $2$-form on $G.$ Then $\omega (e)$ is a $2$-cocycle on $\mathfrak {g}$ i.e. $$\omega (e) ([a,b], c) + \omega(e) ([b,c], a) + \omega (e) ([c,a], b) = 0$$ for all $a,b,c \in \mathfrak {g},$ where $[\cdot, \cdot]$ is the Lie bracket on $\mathfrak {g}.$
How to prove it? Any help in this regard would be warmly appreciated.
Thanks for your time.