IPA: /ɪmˈpɪəriəli/
KK: /ɪmˈpɪrɪəli/
In a way that relates to an empire or is characteristic of an emperor.
The ruler governed imperially, making decisions that affected the entire empire.
Imperially is formed from "imperial" (from Latin "imperialis", meaning of an empire) and "-ly" (meaning in a manner). The word describes something done in the manner of an empire or relating to an empire.
Think of something done 'in the manner of an empire' — 'imperial' relates to an empire, and '-ly' indicates the manner.