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.
Descrition 1: $$ \begin{matrix} \alpha_0&=& (1,-1,0,0,0,0,0,0,0)\\ \alpha_1&=& (0,1,−1,0,0,0,0,0,0)\\ \alpha_2&=& (0,0,1,−1,0,0,0,0,0)\\ \alpha_3&=& (0,0,0,1,−1,0,0,0,0)\\ \alpha_4&=& (0,0,0,0,1,−1,0,0,0)\\ \alpha_5&=& (0,0,0,0,0,1,−1,0,0)\\ \alpha_6&=& (0,0,0,0,0,0,1,−1,0)\\ \alpha_7&=& (0,0,0,0,0,0,0,1,−1)\\ \alpha_8&=& -\tfrac13( 1,1,1,1,1,1,−2,−2,−2)\,\,\,\, \end{matrix} $$
Descrition 2: $$ \begin{matrix} \alpha_0&=& (1,-1,0,0,0,0,0,0)\\ \alpha_1&=& (0,1,−1,0,0,0,0,0)\\ \alpha_2&=& (0,0,1,−1,0,0,0,0)\\ \alpha_3&=& (0,0,0,1,−1,0,0,0)\\ \alpha_4&=& (0,0,0,0,1,−1,0,0)\\ \alpha_5&=& (0,0,0,0,0,1,−1,0)\\ \alpha_6&=& (0,0,0,0,0,0,1,−1)\\ \alpha_7&=&-\tfrac12(1,1,1,1,1,1,1,-1)\,\,\,\,\\ \alpha_8&=& (0,0,0,0,0,0,1,1) \end{matrix} $$
Descrition 3: $$ \begin{matrix} \alpha_0&=& (1,-1,0,0,0)\oplus(0,0,0,0,0)\\ \alpha_1&=& (0,1,−1,0,0)\oplus(0,0,0,0,0)\\ \alpha_2&=& (0,0,1,−1,0)\oplus(0,0,0,0,0)\\ \alpha_3&=& (0,0,0,1,−1)\oplus(0,0,0,0,0)\\ \alpha_4&=& -\tfrac15[(1,1,1,1,−4)\oplus(3,3,−2,−2,−2)]\,\,\,\,\\ \alpha_5&=& (0,0,0,0,0)\oplus(0,1,-1,0,0)\\ \alpha_6&=& (0,0,0,0,0)\oplus(0,0,1,-1,0)\\ \alpha_7&=& (0,0,0,0,0)\oplus(0,0,0,1,-1)\\ \alpha_8&=& (0,0,0,0,0)\oplus(1,-1,0,0,0) \end{matrix} $$
Descrition 4: $$ \begin{matrix} \alpha_0&=& (1,-1,0,0,0,0,0,0)\oplus(0,0)\\ \alpha_1&=& (0,1,-1,0,0,0,0,0)\oplus(0,0)\\ \alpha_2&=& (0,0,1,-1,0,0,0,0)\oplus(0,0)\\ \alpha_3&=& (0,0,0,1,-1,0,0,0)\oplus(0,0)\\ \alpha_4&=& (0,0,0,0,1,-1,0,0)\oplus(0,0)\\ \alpha_5&=& (0,0,0,0,0,1,-1,0)\oplus(0,0)\\ \alpha_6&=& -\tfrac14[(1,1,1,1,1,1,−3,−3)\oplus(2,−2)]\,\,\,\,\\ \alpha_7&=& (0,0,0,0,0,0,0,0)\oplus(1,-1)\\ \alpha_8&=& (0,0,0,0,0,0,1,-1)\oplus(0,0) \end{matrix} $$
Descrition 5: $$ \begin{matrix} \alpha_0&=& (1,-1,0,0,0,0)\oplus(0,0,0)\oplus(0,0)\\ \alpha_1&=& (0,1,-1,0,0,0)\oplus(0,0,0)\oplus(0,0)\\ \alpha_2&=& (0,0,1,-1,0,0)\oplus(0,0,0)\oplus(0,0)\\ \alpha_3&=& (0,0,0,1,-1,0)\oplus(0,0,0)\oplus(0,0)\\ \alpha_4&=& (0,0,0,0,1,-1)\oplus(0,0,0)\oplus(0,0)\\ \alpha_5&=& -\tfrac16[(1,1,1,1,1,−5)\oplus(4,−2,−2)\oplus(3,−3)]\\ \alpha_6&=& (0,0,0,0,0,0)\oplus(1,-1,0)\oplus(0,0)\\ \alpha_7&=& (0,0,0,0,0,0)\oplus(0,1,-1)\oplus(0,0)\\ \alpha_8&=& (0,0,0,0,0,0)\oplus(0,0,0)\oplus(1,-1) \end{matrix} $$
Descrition 6: $$ \begin{matrix} \alpha_0&=& (1,-1,0,0)\oplus(0,0,0,0,0)\\ \alpha_1&=& (0,1,-1,0)\oplus(0,0,0,0,0)\\ \alpha_2&=& (0,0,1,-1)\oplus(0,0,0,0,0)\\ \alpha_3&=& -\tfrac14[(1,1,1,−3)\oplus(2,−2,−2,−2,−2)]\\ \alpha_4&=& (0,0,0,0)\oplus(1,-1,0,0,0)\\ \alpha_5&=& (0,0,0,0)\oplus(0,1,-1,0,0)\\ \alpha_6&=& (0,0,0,0)\oplus(0,0,1,-1,0)\\ \alpha_7&=& (0,0,0,0)\oplus(0,0,0,1,-1)\\ \alpha_8&=& (0,0,0,0)\oplus(-1,-1,0,0,0) \end{matrix} $$