Exercise 14.33 in Fulton and Harris's Representation Theory claims that
If the roots of a semisimple Lie algebra lie in a collection of mutually orthogonal subsets, one sees that the Lie algebra decomposes into a direct sum accordingly.
Although intuitively I can see why this is true, could anybody direct me to a proof of this assertion, or outline how I should go about doing it?
It seems that you only want hints, so here are some:
Hint 1: Compute $(\alpha+\beta,\alpha)$ and $(\alpha+\beta,\beta)$ for $\alpha$, $\beta$ taken from one of the orthogonal subsets.
Hint 1.5: Deduce that $\alpha+\beta$ is not a root.
Hint 2: Use that $[\mathfrak{g}_\alpha,\mathfrak{g}_\beta]=\mathfrak{g}_{\alpha+\beta}$.
Hint 3: Take $\mathfrak{g}^i$ to be the some of all $\mathfrak{g}_\alpha$ in one of the orthogonal subsets.