In this question, the space $E$ has dimension $n \geq 2$. For every pair $(\alpha , \beta)$ of non-zero vectors of $E$, let $\theta _ { \alpha , \beta }$ be the geometric angle between $\alpha$ and $\beta$, that is, the unique element of $[ 0 , \pi ]$ given by: $\| \alpha \| . \| \beta \| \cos \theta _ { \alpha , \beta } = \langle \alpha , \beta \rangle$.

Let $\mathcal { R }$ be a root system of $E$ and let $\alpha , \beta$ be two non-collinear elements of $\mathcal { R }$.

a) Show, using property 4, that: $2 \frac { \| \alpha \| } { \| \beta \| } \left| \cos \theta _ { \alpha , \beta } \right| .2 \frac { \| \beta \| } { \| \alpha \| } \left| \cos \theta _ { \alpha , \beta } \right| \leq 3$.

b) Assume $\| \alpha \| \leq \| \beta \|$. Show that the pair $(\alpha , \beta)$ is found in one of the configurations listed in the table below (each row corresponding to a configuration):
\begin{center}
\begin{tabular}{ | c | c | c | }
\hline
$\theta _ { \alpha , \beta }$ & $\cos \theta _ { \alpha , \beta }$ & $\| \beta \| / \| \alpha \|$ \\
\hline
$\pi / 2$ & 0 & $\geq 1$ \\
$\pi / 3$ & $1 / 2$ & 1 \\
$2 \pi / 3$ & $- 1 / 2$ & 1 \\
$\pi / 4$ & $\sqrt { 2 } / 2$ & $\sqrt { 2 }$ \\
$3 \pi / 4$ & $- \sqrt { 2 } / 2$ & $\sqrt { 2 }$ \\
$\pi / 6$ & $\sqrt { 3 } / 2$ & $\sqrt { 3 }$ \\
$5 \pi / 6$ & $- \sqrt { 3 } / 2$ & $\sqrt { 3 }$ \\
\hline
\end{tabular}
\end{center}