I'm not sure what you're looking for here. I think the definition of imperium you're looking for here is:

4 The exercise of authority, rule, discipline.

From the OLD. The first definition is:

The supreme administrative power, in Rome exercised at first by the kings, and subsequently by certain magistrates and provincial governors.

The shorthand I use in my classes is:

Imperium is the right and power to give orders and see them carried out.

I stole that from a professor.

Imperator is

1 One who gives orders, a person in charge....

2 A commanding officer, general....

3 A title of honour conferred on a victorious general by acclamation of his troops or by a vote of the senate....

The relationship between the two words is pretty straight forward.

Imperium can also have the sense of empire, as in imperium Romanorum. I wrote a little bit about that here.

If we could clarify your question, maybe we could get beyond the dictionary?