Length of root strings is at most 4 Let $\Phi$ be a root system. In his Introduction to Lie algebras and Representation Theory, J. Humphreys proves that if
$$\beta-p\alpha,\dots,\beta,\dots,\beta+q\alpha$$
is the $\alpha$-root string through $\beta$, then $$\frac{2(\beta,\alpha)}{(\alpha,\alpha)}=p-q.$$
Then, he states that it follows "at once" that root strings are of length at most four.
I dont get why. Of course, we know that $-4\le 2(\beta,\alpha)/(\alpha,\alpha)\le 4$, but then length of the root string is $p+q+1$, not $p-q$...
 A: Here is a slight refinement of Sid's argument. Using Humphreys'  notation we know that $-3 \le \langle \alpha,\beta \rangle \le 3$ for all roots $\alpha,\beta$. Hence, in particular, for the $\alpha$-string through $\beta$ of length $q+r+1$ given by $\beta-r\alpha,\ldots ,\beta+q\alpha$ we find
$$
\langle \beta+q\alpha,\alpha\rangle \le 3\mbox{ and} -3\le \langle \beta-r\alpha,\alpha\rangle.
$$
But $\langle \alpha,\alpha\rangle=2$ therefore implies
$$
2q\le 3- \langle \beta ,\alpha\rangle  \mbox{ and } 2r\le 3+\langle \beta,\alpha\rangle.
$$
Thus together we find $q+r+1\le 4$.
A: By positive definiteness $0 \leq (\alpha, \beta)^2 < (\alpha, \alpha)( \beta, \beta )$ or, equivalently, $0 \leq (\alpha, \beta^\vee) (\beta, \alpha^\vee ) < 4$.  Note the strict inequality.
Edit: I replaced $\langle \, , \, \rangle$ by $(\, , \,)$ to denote the inner product because of their conflicting interpretations.
A: I had trouble with exactly the same sentence in Humphrey's book (last sentence before the exercise on page 45 --- there was no "at once" for me).  After googling, I ended up on this webpage, but I found the exchange of remarks tricky to follow.  For the record, an easy way to prove that a root string has length at most 4 is to note that the sequence
 $$ \langle \beta + i \alpha, \alpha \rangle, \enspace \hbox {where $i$ is an integer}, $$
 forms an AP with difference 2.  If $\phi$ and $\psi$ are any two linearly independent roots, then $$ \langle \psi, \phi \rangle =-3, -2, -1, 0, 1, 2 \hbox { or } 3, $$
from which it follows that the root string has length at most 4.
The formula $r - q = \langle \beta, \alpha \rangle$ becomes really useful slightly later when constructing the Hasse diagram of the positive roots from a base of simple roots using root strings:
 $$ \hbox {string length} = r + q + 1 = 2r + 1 - \langle \beta, \alpha \rangle. $$ 
 In particular, when $r=0$ (that is, when $\beta - \alpha$ is not a root, as happens when $\alpha$ and $\beta$ are simple roots), then the string length is $1 - \langle \beta, \alpha \rangle$.
