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\paragraph{\bf Exercise 4:} Some things about roots.
\begin{enumerate}
\item 
\begin{enumerate}
\item Calculate the roots for types $B_r$, $C_r$, and $D_r$.
\item Draw the roots for $B_1, B_2, C_1, C_2$, and $D_2$ (these can all be drawn in one or two dimensions). \\
Note: compare your pictures to your answers for Exercise 1, part (2)!
\end{enumerate}
\item For $\alpha, \beta \in R$,  show that
\begin{enumerate}
	\item $\beta(h_{\alpha^\vee}) \in \ZZ$,
	\item $\beta - \beta(h_{\alpha^\vee})\alpha \in R$, and 
	\item if $\beta \neq \pm \alpha$, and $a$ and $b$ are the largest non-negative integers such that 
	$$\beta - a\alpha \in R \qquad \text{ and } \beta  + b \alpha \in R,$$
	then $\beta + i\alpha \in R$ for all $-a \leq i \leq b$ and $\beta(h_{\alpha^\vee}) = a-b$. 
\end{enumerate}
(Use the fact that $\sum_i \fg_{\beta + i \alpha}$ is a $\fsl_2$-module.)
\end{enumerate}


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