I am learning basic Lie algebra. I want to prove the following (with standard notations): if $\alpha \in \Delta$ is a root, then the dimension of $\mathfrak{g}_\alpha$ is one and $F\alpha \cap \Delta = \{\pm\alpha\}$, where $F$ is the base field. The proof I found goes as follows.
There is a Lie subalgebra $S_\alpha$ isomorphic to $\mathfrak{sl}_2$. Introduce $$M=\mathfrak{h} \oplus \bigoplus_{c \in F^\times} \mathfrak{g}_{c\alpha}$$ which is a $S_\alpha$-submodule (the base elements $x_\alpha$ and $y_\alpha$ raise or lower the weights by $2$, so that it remains in $M$). Now the weights are obviously $0$ and $c\alpha(h_\alpha) = 2c \in \mathbb{Z}$.
Ker $\alpha$ is a subalgebra of codimension $1$ in $\mathfrak{h}$ (weight zero). And now a statement I don't understand: $S_\alpha$ is also weight zero, with dimension 1, and the weights are then only $0$ and $2$. Why is $S_\alpha$ weight zero, and why is the only other weight $2$? (and not $-2$ or $c\alpha$?)
I may be unclear with the precise difference between root and weight.
