Let $\Phi \subset E$ be a finite reduced crystallographic root system in a finite-dimensional Euclidean space $E$, and let $\Delta \subset \Phi$ be a simple root basis. Let $\Gamma(\Delta)$ denote the Dynkin diagram whose vertices are the elements of $\Delta$, with an edge between distinct vertices $\alpha,\beta \in \Delta$ precisely when $(\alpha,\beta)_E \ne 0$. Then $\Gamma(\Delta)$ is connected if and only if $\Phi$ is irreducible, meaning that $\Phi$ cannot be written as a disjoint union $\Phi = \Phi_1 \sqcup \Phi_2$ of two nonempty root subsystems spanning mutually orthogonal subspaces of $E$.