Show that a reducible Coxeter group is crystallographic if and only if each of its irreducible direct factors is crystallographic.
Note: A Group $G$ is said to be Crystallographic if there is a Lattice $\mathfrak L$ (integer linear combinations of n linear independent vectors) invariant under $G$, i.e., $Tx\in\mathfrak L$ for all $T\in G$, all $x\in\mathfrak L$.
Proof.
"$\Rightarrow$" Let $G$ be a reducible crystallographic Coxeter group. So its base is the union of two nonempty orthogonal subsets: $\Pi=\Pi_1\cup\Pi_2$ and we can write $G=G_1\times G_2$ where $G_1,G_2$ are the corresponding Coxeter groups for $\Pi_1,\Pi_2$. Also there is a lattice $\mathfrak L$ invariant under $G$. How do I go on from here? I need to show that there are lattices for $G_1$ and $G_2$ but I have no idea how to obtain them. Can I take $\mathfrak L$ and write it as $\mathfrak L_1 \oplus\mathfrak L_2$ where $\mathfrak L_1$ and $\mathfrak L_2$ can be generated by $\Pi_1$ and $\Pi_2$?
Edit: "$\Leftarrow$" seems to be trivial: We have Lattices $\mathfrak L_1$ for $G_1$ and $\mathfrak L_2$ for $G_2$. So $\mathfrak L_1\oplus\mathfrak L_2$ is a lattice for $G_1\times G_2=G$, so $G$ is crystallographic.