There is no single answer to this question. Here is one observation.
The element $z_G=e^{2\pi i\rho^\vee}$, where $\rho^\vee$ is one-half the sum of the positive co-roots, is a canonical (independent of the choice of positive co-roots) element of $G$, fixed by every automorphism of $G$. If $G$ is simply connected $z_G=1$ if and only if $\rho^\vee$ is in the co-root lattice. For whatever reason this $z_G$ tends to come up. For a slightly more precise list of your cases see Bourbaki, Lie Groups and Lie Algebras, Chapters 7-9, Chapter IX, Section 4, Exercise 13.
For example the Frobenius-Schur indicator of a self-dual finite dimensional representation $V_\lambda$ (telling whether the invariant form is orthogonal or symplectic) is $e^{2\pi i\langle\lambda,\rho^\vee\rangle}=\lambda(z_\rho)$. So $\rho^\vee$ is in the root lattice if and only if every such representation of the simply connected group is orthogonal. See [loc. cit. Chapter 9, Section 7] and Math Overflow: Is there a formula for the Frobenius-Schur indicator of a rep of a Lie group?