  Let $\Lambda_Q$ be the preprojective algebra of a finite acyclic quiver $Q$
of non-Dynkin type and $D^b(\mathrm{rep}^n \Lambda_Q)$ be the bounded derived
category of finite dimensional nilpotent $\Lambda_Q$-modules. We define
spherical twist functors over the root category $\mathcal{R}_{\Lambda_Q}$ of
$D^b(\mathrm{rep}^n \Lambda_Q)$ and then realize the Weyl group associated to
$Q$ as certain subquotient of the automorphism group of the Ringel-Hall Lie
algebra $\mathfrak{g}(\mathcal{R}_{\Lambda_Q})$ of $\mathcal{R}_{\Lambda_Q}$
induced by spherical twist functors. We also present a conjectural relation
between certain Lie subalgebras of $\mathfrak{g}(\mathcal{R}_{\Lambda_Q})$ and
$\mathfrak{g}(\mathcal{R}_Q)$, where $\mathfrak{g}(\mathcal{R}_Q)$ is the
Ringe-Hall Lie algebra associated to the root category $\mathcal{R}_Q$ of $Q$.
