I want to show that a coxeter graph $\Gamma$ is connected if and only if its root system $\Phi$ is irreducible. So let $\Delta$ be a simple system of $\Phi$, and $\Delta$ is also our simple system. Let $W$ be the group generated by the reflections $s_\alpha$ where $\alpha \in \Phi$.
Here's my problem: For one part, I assume $\Gamma$ is not connected, and I want to show that $\Phi$ is reducible. So I write $\Delta = \Delta' \cup \Delta''$ as a disjoint union. Then, set $\Phi' = W\Delta'$ and $\Phi'' = W \Delta''$. Now $\Phi = \Phi' \cup \Phi''$, but why is this a disjoint union?
Let $\langle\cdot\,,\cdot\rangle$ be a $W$-invariant inner product on the real vector space $V$ spanned by $\Delta$. If $\Gamma$ can be split into two subgraphs $\Gamma'$ and $\Gamma''$ such that no vertex in $\Gamma'$ is joined to a vertex in $\Gamma''$, then the corresponding sets of vertices $\Delta'$ and $\Delta''$ of $\Gamma'$ and $\Gamma''$, respectively, are orthogonal to each other with respect to $\langle\cdot\,,\cdot\rangle$. Let $W'$ be the subgroup of $W$ generated by the reflections along the roots $\alpha' \in \Delta'$, and similarly, let $W''$ be the subgroup of $W$ generated by the roots $\alpha'' \in \Delta''$. Since $\langle\alpha',\alpha''\rangle = 0$ whenever $\alpha' \in \Delta'$ and $\alpha'' \in \Delta''$, the reflections $s_{\alpha'}$ and $s_{\alpha''}$ commute. Also, $W'$ fixes $\Delta'$ point-wise, and similarly, $W''$ fixes $\Delta'$ point-wise. Hence, $W = W'W'' = W''W' \simeq W'\times W''$ and therefore, $\Phi' = W\Delta' = W'\Delta'$ and $\Phi'' = W\Delta'' = W''\Delta''$ are root subsystems of $\Phi$ spanning complementary (orthogonal to each other) subspaces of $V$.