In the Book "Representation Theory" by Fulton and Harris, this fact ist stated on page 353 after looking at the weight diagrams of the complex Lie-Algebra $G_2$. The authors deduce that with $V=\Gamma_{1,0} $ being the 7-dimensional standard representation of $G_2$ with heighest weight $\omega_1 =2 \alpha_1 + \alpha_2$: $\Lambda V = \Gamma_{0,1} \oplus V$, with $\Gamma_{0,1}$ being the adjoint representation with highest weight $\omega_2 =3 \alpha _1 + 2 \alpha _2$. They then state the following, which I don't quite see:
"In particular, since the adjoint representation $\Gamma_{0,1}$ of $G_2$ is contained in $\Lambda V$, and the irreducible representation $\Gamma_{a,b}$ with highest weight $a\omega_1 + b\omega_2$ is contained in the tensor product $Sym^a V \otimes Sym^b \Gamma_{0,1}$, we see that every irreducible representation of $G_2$ appears in some tensor power $V^{\otimes m}$ of the standard representation."
It would be nice if someone could shed some light on this for me. Note that I dont have the full book here, I could only copy what chapters I thought I needed, so basically only the chapter about the representations of $G_2$.
Thanks in advance!
Since $\Gamma_{0,1}$ is contained in $\Lambda^2V\subset V\otimes V$, you see that $\Gamma_{a,b}\subset Sym^a V\otimes Sym^b \Gamma_{0,1}\subset \otimes^a V\otimes \otimes^b(\otimes^2 V)\subset\otimes^{a+2b}V$.