In section $22.4$ of "Representation Theory: A First Course" by Fulton and Harris, the exceptional Lie algebra $\mathfrak{e}_8$ is constructed using a method of Freudenthal. For background, I will briefly describe the construction. Let $\mathfrak{g}_0:=\mathfrak{sl}_9$ and let $V$ be the standard $9$-dimensional representation of $\mathfrak{g}_0$. Let $W=\bigwedge^3V$ and consider the vector space
$$\mathfrak{g}=\mathfrak{g}_0\oplus W\oplus W^*$$
The bracket on $\mathfrak{g}$ is then defined using the action of $\mathfrak{g}_0$ on $W$ and $W^*$, the trilinear maps $W^{\otimes 3}\to\mathbb{C}$ and $(W^*)^{\otimes 3}\to\mathbb{C}$ given by the wedge product, and the Killing form on $\mathfrak{g}_0$.
My question concerns part of exercise $22.21$, which is to verify that the Dynkin diagram of $\mathfrak{g}$ is $E_8$. Another result in the same section says that the Cartan subalgebra of $\mathfrak{g}$ is the same as that of $\mathfrak{g}_0$. Hence we may choose the simple roots of $\mathfrak{g}$ to be
$$\{(L_i-L_{i+1},0,0)\mid i=1,2,\ldots,8\}$$
where $L_i(A)=A_{ii}$ for any traceless diagonal matrix $A$. How do I show that these roots give the Dynkin diagram $E_8$? In order to use these roots to construct the Dynkin diagram for $\mathfrak{g}$, I know that I need to be able to compute the Killing form ($\alpha,\beta$) for two simple roots $\alpha$ and $\beta$, but I don't know how to do this.