The root system of $sl(3,\mathbb C)$

The root system of $A_2$ consists of $\Phi=\{ \pm \alpha,\pm \beta , \pm (\alpha+\beta) \}$, which can be constructed as follows. With the Cartan subalgebra $\mathfrak{h}_{\mathbb{R}}\cong \mathbb{R}^2$ and the canonical scalar product on $\mathbb{R}^2$ we can realize the simple roots as $$ \alpha=\begin{pmatrix} -\frac{1}{2} \\ \frac{\sqrt{3}}{2} \end{pmatrix}, \quad \beta=\begin{pmatrix} 1 \\ 0 \end{pmatrix}. $$ Then we obviously have $(\alpha,\alpha)=(\beta,\beta)=1$, and in addition \begin{align*} \langle \alpha, \alpha^{\vee}\rangle & = \frac{2(\alpha,\alpha)}{(\alpha,\alpha)}= 2 ,\\ \langle \alpha, \beta^{\vee}\rangle & = \frac{2(\alpha,\beta)}{(\alpha,\alpha)}= -1,\\ \langle \beta, \beta^{\vee}\rangle & =2, \end{align*} and we obtain the Cartan numbers for type $A_2$. This shows that the root system of $A_2$ consists of $\Phi=\{ \pm \alpha,\pm \beta , \pm (\alpha+\beta) \}$. For additional information also see the MSE question here, for the case of $B_2$. Of course, the same discussion is valid for type $A_2$.