Every irreducible representation of $G_2$ appears in some tensor power of the standard representation 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 $\Gamma_{a,b}\subset Sym^a V\otimes Sym^b \Gamma_{0,1}\subset \otimes^a V\otimes \otimes^b(\otimes^2 V)\subset\otimes^{a+2b}V$.
