questions about coroot

Question

I am reading the lecture notes of geometric representation theory: http://www.math.harvard.edu/~gaitsgde/267y/catO.pdf. I have a question on coroot. In general, if we have a root $\alpha$, then the corresponding coroot is $\check{\alpha}=2\alpha/(\alpha, \alpha)$. But on page 1 of the lecture notes, line -14, it is said that $h_{\check{\alpha}}$ is a coroot. I don't understand this. Further, why $[e_{\alpha}, f_{\alpha}] \neq h_{\alpha}$ but $[e_{\alpha}, f_{\alpha}]$ is proportionate to $h_{\check{\alpha}}$? Thank you very much.

Answer 1

You surely will not get $h_{\alpha^\vee}$ for all choices of $e_\alpha\in\mathfrak n_\alpha$ and $f_\alpha\in\mathfrak n_\alpha^-$, both non-zero, but otherwise arbitrary!

You should read through Jim Humphreys's Introduction to Lie Algebras, which is amazingly readable.