A semi-simple Lie group has a Cartan Subalgebra ($H$) (CSA) -an LVS, Dual to this CSA LVS is root space($H^*$), which is set funtionals that map elements of CSA to real numbers and hence useful in defining inner product. If $\alpha(C_H) \in H^*$ where $C_H \in H$ is arbitrary element in $H$.
It is the Killing form that gives the map between $H$ and $H^*$, taking $K \in H$
$$ (C_H,K) = \alpha(K) \qquad \text{where} \qquad (C_H,K) = \mathrm{Tr}(\mathrm{adj}C_H\mathrm{adj}K) \;\; \text {is the killing form} $$
Now this Root space is supposed to form an LVS. Therefore for every root $\alpha$ we have a root $-\alpha$. In Robert Cahn's book on Semi-simple Lie algebra, he says
"If $\alpha$ is root then $-\alpha$ is also root. Secondly, to every corresponding $\alpha$ there is a generator in the Lie Algebra. Thirdly, if $\alpha$ is a root, $2\alpha$ is not a root".
I am not able to convince myself about the fact why $2\alpha$ is not a root. $H^*$ is an LVS, hence any linear combination should work. Is he trying to refer to the fact that the a root $2\alpha$ will give me a generator $2E_{\alpha} \in \mathcal{L}$, the Lie algebra and so we can't consider it as a giving me some unique root ? Please clarify.