I'm new to lattice groups and I'm stuck in the very first proposition (2.1) of Paul Conrad's paper "Characteristic Subgroups of Lattice-Ordered Groups" (http://www.jstor.org/stable/1995910). Part (c) of the statement says that
"If $A$ is an atom of the lattice of $\ell$-ideals, and it is a summand of $G$, then it is simple; moreover if $G$ is representable, then $A$ is linearly ordered."
When I look at the proof, I don't see where $A$ being a summand or where $G$ being representable comes in. Aren't atoms, almost by definition, simple?