In order to understand $E_8$, it is useful to have many different perspectives. Here are some ways of writing its eight simple roots $\alpha_1,\ldots,\alpha_8$ in various coordinate systems. I'll also list the lowest root $\alpha_0=−2\alpha_1 -3\alpha_2 -4\alpha_3 -5\alpha_4 -6\alpha_5 -4\alpha_6 -2\alpha_7 -3\alpha_8$ in those same coordinates.

Description 1:
$$
\begin{matrix}
\alpha_0&=& (1,-1,0,0,0,0,0,0,0)\\
\alpha_1&=& (0,1,−1,0,0,0,0,0,0)\\
\alpha_2&=& (0,0,1,−1,0,0,0,0,0)\\
\alpha_3&=& (0,0,0,1,−1,0,0,0,0)\\
\alpha_4&=& (0,0,0,0,1,−1,0,0,0)\\
\alpha_5&=& (0,0,0,0,0,1,−1,0,0)\\
\alpha_6&=& (0,0,0,0,0,0,1,−1,0)\\
\alpha_7&=& (0,0,0,0,0,0,0,1,−1)\\
\alpha_8&=& -\tfrac13( 1,1,1,1,1,1,−2,−2,−2)\,\,\,\,
\end{matrix}
$$

Description 2:
$$
\begin{matrix}
\alpha_0&=& (1,-1,0,0,0,0,0,0)\\
\alpha_1&=& (0,1,−1,0,0,0,0,0)\\
\alpha_2&=& (0,0,1,−1,0,0,0,0)\\
\alpha_3&=& (0,0,0,1,−1,0,0,0)\\
\alpha_4&=& (0,0,0,0,1,−1,0,0)\\
\alpha_5&=& (0,0,0,0,0,1,−1,0)\\
\alpha_6&=& (0,0,0,0,0,0,1,−1)\\
\alpha_7&=&-\tfrac12(1,1,1,1,1,1,1,-1)\,\,\,\,\\
\alpha_8&=& (0,0,0,0,0,0,1,1)
\end{matrix}
$$

Description 3:
$$
\begin{matrix}
\alpha_0&=& (1,-1,0,0,0)\oplus(0,0,0,0,0)\\
\alpha_1&=& (0,1,−1,0,0)\oplus(0,0,0,0,0)\\
\alpha_2&=& (0,0,1,−1,0)\oplus(0,0,0,0,0)\\
\alpha_3&=& (0,0,0,1,−1)\oplus(0,0,0,0,0)\\
\alpha_4&=& -\tfrac15[(1,1,1,1,−4)\oplus(3,3,−2,−2,−2)]\,\,\,\,\\
\alpha_5&=& (0,0,0,0,0)\oplus(0,1,-1,0,0)\\
\alpha_6&=& (0,0,0,0,0)\oplus(0,0,1,-1,0)\\
\alpha_7&=& (0,0,0,0,0)\oplus(0,0,0,1,-1)\\
\alpha_8&=& (0,0,0,0,0)\oplus(1,-1,0,0,0)
\end{matrix}
$$

Description 4:
$$
\begin{matrix}
\alpha_0&=& (1,-1,0,0,0,0,0,0)\oplus(0,0)\\
\alpha_1&=& (0,1,-1,0,0,0,0,0)\oplus(0,0)\\
\alpha_2&=& (0,0,1,-1,0,0,0,0)\oplus(0,0)\\
\alpha_3&=& (0,0,0,1,-1,0,0,0)\oplus(0,0)\\
\alpha_4&=& (0,0,0,0,1,-1,0,0)\oplus(0,0)\\
\alpha_5&=& (0,0,0,0,0,1,-1,0)\oplus(0,0)\\
\alpha_6&=& -\tfrac14[(1,1,1,1,1,1,−3,−3)\oplus(2,−2)]\,\,\,\,\\
\alpha_7&=& (0,0,0,0,0,0,0,0)\oplus(1,-1)\\
\alpha_8&=& (0,0,0,0,0,0,1,-1)\oplus(0,0)
\end{matrix}
$$

Description 5:
$$
\begin{matrix}
\alpha_0&=& (1,-1,0,0,0,0)\oplus(0,0,0)\oplus(0,0)\\
\alpha_1&=& (0,1,-1,0,0,0)\oplus(0,0,0)\oplus(0,0)\\
\alpha_2&=& (0,0,1,-1,0,0)\oplus(0,0,0)\oplus(0,0)\\
\alpha_3&=& (0,0,0,1,-1,0)\oplus(0,0,0)\oplus(0,0)\\
\alpha_4&=& (0,0,0,0,1,-1)\oplus(0,0,0)\oplus(0,0)\\
\alpha_5&=& -\tfrac16[(1,1,1,1,1,−5)\oplus(4,−2,−2)\oplus(3,−3)]\\
\alpha_6&=& (0,0,0,0,0,0)\oplus(1,-1,0)\oplus(0,0)\\
\alpha_7&=& (0,0,0,0,0,0)\oplus(0,1,-1)\oplus(0,0)\\
\alpha_8&=& (0,0,0,0,0,0)\oplus(0,0,0)\oplus(1,-1)
\end{matrix}
$$

Description 6:
$$
\begin{matrix}
\alpha_0&=& (1,-1,0,0)\oplus(0,0,0,0,0)\\
\alpha_1&=& (0,1,-1,0)\oplus(0,0,0,0,0)\\
\alpha_2&=& (0,0,1,-1)\oplus(0,0,0,0,0)\\
\alpha_3&=& -\tfrac14[(1,1,1,−3)\oplus(2,−2,−2,−2,−2)]\\
\alpha_4&=& (0,0,0,0)\oplus(1,-1,0,0,0)\\
\alpha_5&=& (0,0,0,0)\oplus(0,1,-1,0,0)\\
\alpha_6&=& (0,0,0,0)\oplus(0,0,1,-1,0)\\
\alpha_7&=& (0,0,0,0)\oplus(0,0,0,1,-1)\\
\alpha_8&=& (0,0,0,0)\oplus(-1,-1,0,0,0)
\end{matrix}
$$