= Solution
A <Weyl chamber> is a connected component of
$$
E\setminus\bigcup_{\alpha\in\Phi}\alpha^\perp.
$$
A <root basis> $\Delta\subseteq\Phi$ is a vector-space basis of $E$ such that every root is an integer combination of elements of $\Delta$ with all nonzero coefficients of one sign.
To construct one, choose a regular vector $\gamma\in E$, meaning $(\gamma,\alpha)\ne0$ for every root. Declare
$$
\Phi_\gamma^+=\{\alpha\in\Phi:(\gamma,\alpha)>0\}.
$$
The positive roots in $\Phi_\gamma^+$ which cannot be written as sums of two positive roots form a root basis $\Delta_\gamma$. Vectors $\gamma$ in the same Weyl chamber give the same basis.
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