Currently I'm reading Representations of Compact Lie Groups (Brocker) and in V-2.1 Brocker writes a decomposition of $\mathfrak{g}$ and $\mathfrak{g_\mathbb{C}}$ by the weight spaces. The book marks:
$$L_\chi = \{v\in \mathfrak{g}_\mathbb{C}\mid \forall_{h\in{\mathfrak{t}_\mathbb{C}}} [h,v]= \chi(h)\cdot v \}$$
Where $\chi$ is some complex root of the torus (lie algebra). Then the book says that we can decompose:
$$ \mathfrak{g}_\mathbb{C} = L_0 \oplus \bigoplus_{\chi \in R} L_\chi$$
where $R$ is all of the real roots of $\mathfrak{t}$. Now my first question is why is this decomposition even true? I would understand if $R$ is the set of complex roots (then it's just the regular weight space decomposition) but that's not the case here. Is there a mistake in the book?
My second question involves the second part shown in the book. Here we define: $M_\chi = (L_\chi \oplus L_{-\chi}) \cap \mathfrak{g}$ and than the book claims (without proof) that:
$$\mathfrak{g} = M_0 \oplus \bigoplus_{\chi \in R^+} M_\chi$$
where $R^+$ is a set containing exactly one copy from each pair of real roots $\{\chi, -\chi\}$. Why is this decomposition true?
If I assume that in the decomposition $L_\chi$ is the weight space of the complex root that matches the real root $\chi$, than I was able to show that:
$$ V(\chi) \otimes \mathbb{C} = L_\chi $$
where $V(\chi)$ is the weight space of the real root. Here I got stuck.