Remark 8

Let $R_1, \ldots, R_p$ be root systems in vector spaces $V_1, \ldots, V_p$, $R$ the direct sum of the $R_i$ in $V = \prod_i V_i$, $C_i$ a chamber of $R_i$, and $B_i = B(C_i)$. It is immediate that $C = \prod_i C_i$ is a chamber of $R$ and that $B(C) = \bigcup_i B_i$. It follows from Th. 2 that all the chambers and bases of $R$ are obtained in this way.