How about survival?

If Octavian wanted to be Caesar's heir and live, his best bet was to get rid of Caesar's assassins. Especially if he wanted to inherit not only Caesar's name and fortune but also a political position, and a political position, even if it could not be explicitly mentioned in a will, was surely in everyone's mind.

Survival was what Atia and Philippus were thinking of when they advised the young man not to accept the inheritance.

In the end, Octavian got rid not only of Caesar's assassins but of all his own political ennemies. Love or grief had nothing to do with it.