I am reading the 5the chapter of J-P. Serre's Complex Semisimple Lie Algebras book and I don't understand the following proof given:
Why can we easy conclude that with the fact that $W$ is finite? Also why is the form $$(x,y)= \sum_{w \in W} \; B(wx,wy)$$ invariant? There are some steps that he doesn't explain, which makes it hard for me to understand. Can someone maybe helps me?
You are using two facts in here: the first one is that it is finite to make sure the sum you put makes sense. After that, you sum all possible values because in that way you force invariantness.
More precisely, you know that for a group, as is $W$, the map $w\mapsto ww_0$ is always a bijection, for any $w_0\in W$. Hence \begin{align*} (w_0x, w_0y) &= \displaystyle\sum_{w\in W} (ww_0x, ww_0y)\\ &= \displaystyle\sum_{w\in W} (wx, wy). \end{align*} Furthermore, if $x = y$ then $(x, x)$ becomes the sum of $(wx, wx)\ge 0$, all positive terms. Hence it is positive itself, and it will be $0$ only if all are zero, meaning $wx = 0$ for all $w$, in particular $w = 1$, giving $x = 0$. So, it is indeed positive definite.