Proposition 6

Every root system R in V is the direct sum of a family $(R_i)_{i \in I}$ of irreducible root systems that is unique up to a bijection of the index set.

The existence of the $R_i$ is proved by induction on Card R: if R is non-empty and not irreducible, R is the direct sum of two root systems $R', R''$ such that Card $R' <$ Card $R$, Card $R'' <$ Card $R$, and the induction hypothesis applies to $R'$ and $R''$. To prove uniqueness, it suffices to prove that, if $R$ is the direct sum of $R'$ and $R''$, every $R_i$ is necessarily contained either in $R'$ or in $R''$. Let $V', V'', V'_i, V''_i$ be the vector subspaces of $V$ generated by $R', R'', R' \cap R_i, R'' \cap R_i$. Since the sum $V' + V''$ is direct, the sum $V'_i + V''_i$ is direct. Since $R_i \subseteq R' \cup R''$, $R_i$ is the direct sum of the root systems $R' \cap R_i$ and $R'' \cap R_i$; hence either $R' \cap R_i = \varnothing$ or $R'' \cap R_i = \varnothing$, which proves the assertion.

The $R_i$ are called the irreducible components of $R$. For any non-zero scalars $\lambda_i$, the union of the $\lambda_i R_i$ is a root system in $V$, whose inverse system is the union of the $\lambda_i^{-1} R_i$, and whose Weyl group is $W(R)$.