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Can somebody please explain this to me?

(ad $\ y)^4(z)=0 $, since root strings have length at most 4.

Note: y and z are root vectors belonging to two negative roots.

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Some context might be helpful. In fact, this is literally from Humphreys, page 116 – wildildildlife Jul 10 '11 at 22:29
up vote 1 down vote accepted

The proof of the statement is given page $45$ of Humphreys as an application of the the Lemma that given two roots $\alpha$ and $\beta$, then $(\alpha,\beta)<0$ implies $\alpha-\beta$ is a root, and $(\alpha,\beta)>0$ implies $\alpha+\beta$ is a root.

Note that if $y$ is a root vector for $\alpha$ and $z$ is a root vector for $\beta$, then $(ad\ y)^k(z)$ is a root vector for $\beta+k\alpha$. Hence, you are interested in a bound for unbroken maximal strings of non-zero roots in $\{\beta+i\alpha\}_{i\in\mathbb Z}$ (in the sense that if $z$ is a vector on one end of the string, then applying $ad\ (y)$ to $z$ repeatedly will get you to the other end of the string.

The Lemma is relevant in showing that the maximal subset of non-zero roots in $\{\beta+i\alpha\}_{i\in\mathbb Z}$ is an unbroken string in the above, and hence it equals $\{\beta-r\alpha,\beta-(r-1)\alpha,\dots,\beta-\alpha,\beta,\beta+\alpha,\dots,\beta+q\alpha\}$. Then this unbroken string corresponds to an irreducible representation of a copy of $\mathfrak sl_2$, which in this case conists of $\mathfrak g_\alpha\oplus\mathfrak g_{-\alpha}\oplus[\mathfrak g_\alpha,\mathfrak g_{-\alpha}]$ where $\mathfrak g_\alpha$ is the root space of $\alpha$, must be invariant under the appropriate reflection by $\sigma_\alpha$, given in this case by $\sigma_\alpha(\gamma)=\gamma-\left<\gamma,\alpha\right>\alpha$, where $\left<\beta,\alpha\right>=\frac{2(\beta,\alpha)}{(\alpha,\alpha)}$ with $(\beta,\alpha)$ being the (dual) Killing form on the dual space of the Cartan algebra.

Invariance under the reflection implies $\sigma_\alpha(\beta+q\alpha)=\beta-r\alpha$, while $\sigma_\alpha(\beta+q\alpha)=\sigma_\alpha(\beta)-q\sigma_\alpha(\alpha)=\beta-\left<\beta,\alpha\right>\alpha-r(\alpha-\left<\alpha,\alpha\right>\alpha)=\beta-\left<\beta,\alpha\right>-q\alpha$, it follows that $\left<\beta,\alpha\right>$=r-q$.

The length of our sequence is $|r-q|+1$, and because on the one hand $\left<\beta,\alpha\right>=\frac{2(\beta,\alpha)}{(\alpha,\alpha)}=\beta(H_\alpha)$ (by properties of the Killing form) is an integer (in general weights are integer-valued on the distingusihed elements $H_\alpha\in\mathfrak g_\alpha\oplus\mathfrak g_{-\alpha}\oplus[\mathfrak g_\alpha,\mathfrak g_{-\alpha}]\cong\mathfrak sl_2$), and on the other $\left<\beta,\alpha\right>\left<\alpha,\beta\right>=4\cos(\theta)^2$ where $\theta$ is the angle between $\alpha$ and $\beta$ according to the Killing form, we may deduce that when $\alpha\neq\pm\beta$ (so $\theta\neq0,\pi)$ $|\left<\beta,\alpha\right>|$ divides the possible integer values $0$,$1$,$2$, or $3$ for $4\cos(\theta)^2$.

Hence, the length $|r-q|+1=|\left<\beta,\alpha\right>|+1$ is at most $4$.

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You should have seen the following: If $z$ belongs to root $\beta$ and $y$ to root $\alpha$, then $(\mathrm{ad}\; y)z$ (when non-zero) belongs to the root $\alpha+\beta$.

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