After the unilateral declaration of independence of Brazil in 1822, breaking up the United Kingdom of Portugal, Brazil and the Algarves, there was a sour but peaceful war between Portugal and Brazil, distance oblige.
Officially, the statute of Brazil as part of the United Kingdom only ended with the Treaty of Rio de Janeiro of 1825, according to which King D.Jo„o VI assumed the title of and dignity of de jure titular Emperor of Brazil, but abdicated that crown to his son D.Pedro, de jure still crown-prince of Portugal, Brazil and the Algarves and, de factu, Emperor of Brazil.
Over a period of time, Austria and Hungary were united in what some called the dual monarchy or the Austro-Hungarian Empire. When one thinks of the word empire, we usually think of the large empires of England, France and Spain- a "mother" country along with overseas territories. Within Europe, what were considered empires in the dual monarchy sense?
I don't think that "we" usually define empires as a country with overseas territories. As you know, there once was a huge Roman Empire that ended in 476 when the last Emperor, Romulus Augustulus was deposed by Germanic forces.
In 962 the Holy Roman Empire of the German Nation was founded. It lasted till 1806, when the Empire was dissolved by the French Emperor Napolťon I. The Empire's last Emperor Franz II had already in 1804 declared himself Emperor Franz I of Austria. The Austo-Hungarian Empire lasted till 1918.
In the 19th century there were several attempts to unite the many German states, but Austria and Prussia were rivals. In 1871, the northern German states were united in one country with the King of Prussia as its Emperor. That Empire also ended in 1918.
Under Adolf Hitler, Germany was called Deutsches Reich and for the last years GroŖdeutsches Reich. But many Nazis called it Das dritte Reich, counting the Holy Roman Empire of the German Nation as the first and the Empire founded by Prussian as the second Reich.