Proposition 7

Let $R$ be a root system in $V$, $(R_i)$ the family of its irreducible components, $V_i$ the vector subspace of $V$ generated by $R_i$, $B$ the invariant symmetric bilinear form on $V$ defined in Prop. 3, and $B'$ a symmetric bilinear form on $V$ invariant under $W(R)$. Then the $V_i$ are pairwise orthogonal with respect to $B'$, and, for all $i$, the restrictions of $B$ and $B'$ to $V_i$ are proportional.

If $v_i \in V_i, v_j \in V_j, i \neq j$, and if $w \in W(R_j)$, then

which shows that $w(v_j) - v_j$ is orthogonal to $v_i$ with respect to $B'$. Since $V_j$ is irreducible for $W(R_j)$, it is generated by the $w(v_j) - v_j$, and it is therefore orthogonal to $V_i$.

The fact that the restrictions of $B$ and $B'$ to each of the $V_i$ are proportional follows from Prop. 1 of Chap. V, § 2, no. 1.