Archduchy

the domain of an archduke or an archduchess.
Historical Examples

The archduchy of Austria was, and Hungary was not, German, in the reign of the emperors.
Down the Rhine Oliver Optic

He also ordered the Jews to be expelled from the archduchy of Austria within six months.
History of the Jews, Vol. IV (of VI) Heinrich Graetz

The revenues of the archduchy were some four millions of dollars.
The Empire of Austria; Its Rise and Present Power John S. C. Abbott

Meanwhile Styria and Carinthia Austria created an archduchy.
Encyclopaedia Britannica, 11th Edition, Volume 3, Part 1, Slice 1 Various

Transleithan, trans-lī′than, adj. beyond the Leitha, the boundary river between the archduchy of Austria and Hungary.
Chambers’s Twentieth Century Dictionary (part 4 of 4: S-Z and supplements) Various

noun (pl) -duchies
the territory ruled by an archduke or archduchess