Example (1)

Let $A$ and $B$ be two ordered groups; let $A \times B$ be given the lexicographic order (i.e. "$(a, b) \leq (a', b')$" is equivalent to "$(a < a') \text{ or } (a = a' \text{ and } b \leq b')$"). The second factor $B$ of $A \times B$ is then, as is seen immediately, an isolated subgroup of $A \times B$.