On axiomatic definition of affine root systems

Question

In Macdonald's book Affine Hecke Algebras and Orthogonal Polynomials, chapter 1 introduces affine root systems. I will recall the definition here: Let $E$ be a non-zero real Euclidean space (finite dimensional), $V$ be its vector space of translations, $F$ is the space of affine-linear functions $E \to E$. An affine root system on $E$ is a subset $S$ of $F$ containing non-constant functions satisfying the following:

1. $S$ spans $F$
2. $s_a(S) \subseteq S$ for all $a \in S$, where $s_a: F \to F$ is the reflection in the hyperplane orthogonal to $a$
3. $\langle S^\vee, S \rangle \subseteq \mathbb{Z}$
4. the (affine) Weyl group $W$ acts properly on $E$, where $W = \langle s_a: a \in S \rangle $

The first three axioms are natural analogues of finite root system definition. Why is the fourth axiom required?