Edited remark about coroots.

semisimple_Lie_algebras/cartan_subalgebras.tex

@@ -251,14 +251,13 @@ \subsection{Properties of the root space decomposition}
 
 
 \begin{rem}
- Let $\alpha \in \Phi$. Then $\alpha^\vee = 2 t_\alpha / \kappa(t_\alpha, t_\alpha)$ because
+ Let $\alpha \in \Phi$. The existence and uniqueness of $\alpha^\vee$ follows from the fact that $[\g_\alpha, \g_{-\alpha}]$ is one-dimensional with $\alpha([\g_\alpha, \g_{-\alpha}]) = k \alpha(t_\alpha) \neq 0$. Also notice that $\alpha^\vee = 2 t_\alpha / \kappa(t_\alpha, t_\alpha)$ because
  \[
   \alpha\left( \frac{2 t_\alpha}{\kappa(t_\alpha, t_\alpha)} \right)
   = \frac{2\alpha(t_\alpha)}{\kappa(t_\alpha, t_\alpha)}
   = \frac{2\kappa(t_\alpha,t_\alpha)}{\kappa(t_\alpha, t_\alpha)}
   = 2.
- \]
-
+ \] 
 \end{rem}
 
 
@@ -280,5 +279,6 @@ \subsection{Properties of the root space decomposition}
 
 
 
+