Exercise 1, § 1

Let R be a root system, $R = R_1 \cup R_2$ a partition of R. Assume that, if $x, y$ are two elements of $R_i$ and if $x + y$ (resp. $x - y$) is a root, then $x + y \in R_i$ (resp. $x - y \in R_i$), where $i = 1, 2$. Then, R is the direct sum of $R_1$ and $R_2$. (By using the Cor. of Th. 1, no. 3, show that if $x \in R_1$ and $y \in R_2$ then $(x|y) = 0.$)