Relation between semisimple Lie Algebras and Killing form

Question

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

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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 $$[X_{-\alpha},[X_{\alpha},X_\beta]] = q(p+1)\frac{\langle\alpha,\alpha\rangle}{2}X_\beta.$$

Answer 1

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 $p=0$.