There are several possibilities how one can write simple roots in terms of coordinates. Firstly, they depend on the numbering of the nodes in the Dynkin diagram. Let's fix the choice for $E_6$ to be
Unfortunately there are still several possibilities. For example, this paper quotes at the second page very different simple roots than Wikipedia, which claims that the simple roots in coordinates are the rows of the following matrix:
Which set is correct? Or are both? My problem is that I get very different result if I use these two different sets. Is there some good resource that lists the different possibilities?
It is straightforward to verify $\langle v,v\rangle = 2$ for all rows $v$ and $\langle v,w\rangle=-1$ precisely for rows joined by a line and $\langle v,w\rangle=0$ otherwise.