I am working on Exercise 10.7 of Erdmann & Wildon's "Introduction to Lie Algebras":
Suppose $L$ is semisimple of dimension $6$. Let $H$ be a Cartan subalgebra of $L$ and let $\Phi$ be the associated set of roots.
(i) Show that $\operatorname{dim}H=2$ and that if $\alpha,\beta\in\Phi$ span $H^{\star}$ then $\Phi=\{\pm\alpha,\pm\beta\}$.
(ii) Hence show that $$[L_\alpha,L_{\pm\beta}]=0~\text{and}~[L_{\pm\beta},[L_{\alpha},L_{-\alpha}]]=0$$ and deduce that the subalgebra $L_{\alpha}\oplus L_{-\alpha}\oplus[L_{\alpha},L_{-\alpha}]$ is an ideal of $L$. Show that $L$ is isomorphic to the direct sum of two copies of $\mathfrak{sl}(2,\mathbf{C})$.
I am able to show (i). For $[L_\alpha,L_{\pm\beta}]=0$, I believe it can be done by showing that if $\alpha+\beta\in\Phi$, then $\operatorname{Span}\{\alpha,\beta\}$ has dimension $1$, thus it is impossible that they span $H^{\star}$. This gives $[L_\alpha,L_\beta]\subseteq L_{\alpha+\beta}=0$.
Yet I have no idea how to do the rest of the problem. For the second part of (ii), I tried to show $[e_{\beta},[e_{\alpha},f_{\alpha}]]=0$ which is equivalent to $$0=[e_\beta,h_\alpha]=-[h_\alpha,e_\beta]=-\beta(h_\alpha)e_\beta=-\frac{2(\beta,\alpha)}{(\alpha,\alpha)}e_\beta,$$ but then this means $(\alpha,\beta)=0$. Is it in general true that $\alpha,\beta$ are orthogonal? If so, how can I show this? I'm pretty sure this has to do with $\operatorname{dim}H=2$, but I couldn't work out the details here.
The book has not covered the classification of root systems yet, so it would be nice if a solution based purely on the properties of root space decomposition is given. Thanks!