# Relation between semisimple Lie Algebras and Killing form

I'm studying Lie Algebras and I encountered this problem that I have no idea how to solve it. Can anyone help me?

Let $$\mathfrak{g}$$ be a semisimple Lie Algebra, and $$\mathfrak{h}$$ a Cartan subalgebra of $$\mathfrak{g}$$. Writing the representation $$\rho: \mathfrak{h}\to \mathfrak{gl(g)}$$ $$\ \ \ \ \ \ \ \ X \mapsto \text{ad}(X)$$ we can decompose $$\mathfrak{g}$$ as $$\mathfrak{g}= \mathfrak{h}\oplus \mathfrak{g}_{\alpha_1} \oplus \ldots \oplus\mathfrak{g}_{\alpha_n},$$ where $$\alpha_1, ...,\alpha_n$$ are the weights of the representation $$\rho$$.

Let $$\langle \cdot, \cdot \rangle$$ be the Killing Form, and $$\alpha,\beta$$ roots of $$\rho$$ such that

$$\kappa_{\alpha,\beta} = \frac{2 \langle \beta,\alpha\rangle}{\langle \alpha,\alpha\rangle} = p-q$$

Problem: If $$\alpha, \beta$$ are simple roots of $$\rho$$, $$X_\alpha \in \mathfrak{g}_\alpha$$, $$X_{-\alpha} \in \mathfrak{g}_{-\alpha}$$ such that $$\langle X_\alpha , X_{-\alpha}\rangle = 1$$ and $$X_\beta \in \mathfrak{g}_{\beta}$$, then $$[X_{-\alpha},[X_{\alpha},X_\beta]] = q(p+1)\frac{\langle\alpha,\alpha\rangle}{2}X_\beta.$$

Hint: Recall that $$X_\alpha$$, $$X_{-\alpha}$$ and $$[X_\alpha,X_{-\alpha}]$$ span a Lie subalgebra of $$\mathfrak g$$ that is isomorphic to $$\mathfrak{sl}(2,\mathbb C)$$. Now given another root $$\beta$$, you have the $$\beta$$-string through $$\alpha$$, which defines a representation of this subalgebra that you can analyze using the representation theory of $$\mathfrak{sl}(2,\mathbb C)$$. If you assume that $$\alpha$$ and $$\beta$$ are both simple, then there are some simplifications, for example, you automatically get $$p=0$$.