The root system of $sl(3,\mathbb C)$ I want to determine the root-system of the lie algebra $sl(3,\mathbb C)$. Does someone know a good (and complete) reference for this problem?
I know that the root-system is $A_2$ but I want to see a complete proof (calculation) for this.
Thanks in advance.
 A: 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 $A_2$.
