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. $a+b$ is a root and $a+2b$ is not a root?

2. $a+2b$ is a root and $a+3b$ is not a root?

3. $a+3b$ is a root?

Thanks!

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!

Let $a,b$ be roots ($a\ne \pm b$) of a Lie algebra 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 is the biggest scenery X such that:

1. $a+b$ is a root and $a+2b$ is not a root?

2. $a+2b$ is a root and $a+3b$ is not a root?

3. $a+3b$ is a root?

Thanks!

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. $a+b$ is a root and $a+2b$ is not a root?

2. $a+2b$ is a root and $a+3b$ is not a root?

3. $a+3b$ is a root?

Thanks!