Let $a,b$ be roots ($a\ne \pm b$) of a Lie algebra $g$ of type $X$, where $X$ can be classic or exceptional $(A,B,C,D,E,F,G)$. It is well known that the length of an $a$-string through $b$ is at most 4.


What are all types of $g$ such that:


1) $a+b$ can be a root and $a+2b$ is not a root?


2) $a+2b$ can be a root and $a+3b$ is not a root?


3) $a+3b$ can be a root?


Thanks!



Lia algebra strings
3) is only possible if $a$ is 3 times longer than $b$, so it only happens in $G\_2$. 


2) is only possible if $a$ is 2 times longer than $b$ (or if $a=b+a'$ for $a'$ a root 3 times longer), so it happens in all non-simply laced types ($B,C,F$ and $G$).


1) happens in every simple root system other than $A\_1$, since this happens whenever $a$ and $b$ are the same length and not orthogonal. 


