Because of their uniqueness as an ethnic group were they considered Germans, Slavs, or something different? Were there any moves to bring them into the Empire away from Austro-Hungary?
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Summary: Bohemians were seen mainly as a Slav ethnic group, although a large minority of Bohemians were Germans or Czechs preferring the German language in official business or high society contact.
The idea of bringing the Lands of the Bohemian Crown into the future unified Germany was considered around 1866 but rejected as not worth pursuing at the time.
First; let's talk Czech cultural identity up to the 1860-ies.
The Lands of the Bohemian Crown were gradually forcibly recatholized since as early as the first years of the 30-year war. These efforts were met with various levels of resistance throughout the 17th century, from lukewarm hollow acceptance of catholicism to small armed rebellions. The Czech language was to be eradicated from noble circles, institutions, etc. Peasants in the countryside, often illiterate, were left alone to talk Czech freely. It wasn't until the 4th quarter of the 18th century when Czech scholars started to form a significant initiative in order to defend the Czech language. Thus the 1st phase of the Czech National Revival begun.
Starting the 19th century it gradually grew in strength and aimed at returning the Czech language to educated circles in cities which at the time already spoke mainly German.
Josef Dobrovský published his Czech grammar book in 1809. Josef Jungmann published the five-volume Czech-German dictionary in the 30-ies. His devotion to the Czech language brought in many neologisms or loanwords from other Slavic languages; he reformed the Czech language to serve any complex usage, including rich poetry, scientific terminology, etc. Other Czech scholars spread reformed Czech into their respective fields.
By the revolutionary year 1848 all the cultural goals of the Revival were pretty much reached. All that was left to do was to pursue further educational, political and scientific goals using the Czech language. In the aftermath of the 1848 revolution though, prominent Czech political figures were suppressed in their activities, some even forced to exile. During the following era of minister of interior Alexander von Bach (1851-1859) people still had limited legal options how to meet in larger numbers to discuss and pursue a common goal. The secret police was effectively organized, paid informers were common. People were somewhat safe to discuss ideas only in pubs.
The fall of minister Bach, followed by October Diploma (1860) and February Patent (1861) brought an end to absolutism in the Austrian Empire and along with it a more relaxed atmosphere in the society. People were now allowed to gather in large numbers and organize themselves for various activities. In Bohemia specifically, because its political elite was suppressed since 1848 and political scene fragmented, people turned their focus on cultural and science areas instead. Countless Czech associations and clubs emerged in the 60-ies that promoted nationalistic ideals. These organizations were typically financially self-sufficient and weren't therefore supervised by the Empire.
Notable organizations included: Sokol (Czech for "falcon") movement (1862), the Czech Physicians Association (1862), the Merkur (Czech for "Mercury") business association (1862), the Czech Lawyers Association (1864), the Association of Engineers and Architects (1865), the American Club of Czech Ladies (1865), the Včela (Czech for "bee") labourer collective (1865).
Probably the most important one was Sokol, a huge organization for men, women and minors, somewhat comparable to nowadays Scout organization. It promoted physical fitness with regular exercising as well as engagement in intellectual activities and upholding high moral values; for both men and women. That was pretty revolutionary at that time since conservative circles saw exercising for women as exhausting or even unhealthy. Before Sokol, women's traditional education was limited to languages, arts and etiquette.
Women of the American Club of Czech Ladies attended Czech lectures on various topics from general educational areas to literature, astronomy, etc. and learned about all the cool modern inventions like fridges, washing machines, sewing machines, etc.
Let's talk the 19th century Central European political scene now, up to the German unification of 1871. Napoleon I. brought an end to the Holy Roman Empire when he created the Confederation of the Rhine in 1806. Neither Prussia nor the Austrian Empire joined this short-lived formation. Nevertheless, after Napoleon's defeat, the Rhine Confederation was replaced by the German Confederation in 1815. This time, the western part of Prussia and parts of the Austrian Empire became members, including the Lands of the Bohemian Crown. Since both Prussia and Austria wanted to be the dominant force within the German Confederation, tension soon emerged between the two powers. In the 60-ies the situation was favourable for Prussia to escalate the tension into a military conflict; the Austrian Empire was isolated with no powerful allies and their recent disagreement over the fate of the Schleswig-Holstein territory served Prussians as a justified means to question the meaning of the German Confederation and provoke the Austrian Empire into war. This dispute finally led to the Austro-Prussian War of 1866 which mainly took place in the Bohemian Crown Lands.
Were Czechs loyal to the Austrian Empire in this conflict? Yes, they were, politicians and peasants alike. The Austrian army even had to reject a romantic offer of Czech students to form a volunteer academic legion. In fact, Czechs were loyal throughout the whole 19th century, including Napoleonic wars. Czechs desired recognition as a nation in the first place, a respectable political representation within the Austrian Empire. They would welcome eg. a federation and not necessarily a separate state. So while the idea to reestablish the Bohemian Kingdom was definitely evaluated by Prussian leaders, they realized the pro-separatist sentiment wasn't really strong and therefore not worth any big effort; some small effort wouldn't hurt though which resulted in a moderate poster/leaflet campaign promoting Bohemian sovereignty and Prussia hopefully playing a positive role in it. Following the decisive Prussian victory at the battle of Hradec Králové and Sadová (Königgrätz and Sadowa) the Prussian army was ordered not to pillage and to generally behave. That meant a chance for friendly relations with Prussia should Bohemia ever really regain sovereignty but realistically, Prussians were simply fine with the 72 days lasting occupation going as smooth as possible without further escalation so that diplomatic talks could end the war with reasonable gains for Prussia.
Prussian demands were reasonable. After all, the most important goal of Austrian expulsion from the German Confederation and the confirmation of Prussia as the leader of the German states was already confirmed on the battlefield. Forcing Austria to secede some of her big core territories (eg. the Bohemian Crown Lands) would arguably drag other European powers into the conflict and hinder the transformation of the now dissolved German Confederation into the new North German Confederation formed in 1867 that consisted of German states north of the river Main.
The Austrian Empire, although defeated and expelled, took the opportunity to reform itself too. In 1867 it transformed itself into Austria-Hungary. That was a very bitter pill to swallow for Czech nationalists as they hoped their loyalty throughout long decades would be rewarded through some kind of political ascension. Yet it was rather a matter of prestige and not personal freedoms. The 1867 constitution of the Austria-governed half declared, among other things, all its nations equal, the right to study in one's native language, the freedom of religion, the right to reach any rank in a state institution, the right to pursue science, etc. and of all its 60 parliament deputies there were guaranteed 15 Czech members. The Hungarian-governed half had another 60 deputies.
Czechs kept establishing more and more associations throughout 70-ies and 80-ies, including: the Historical Club (1872), the Polygraphic Association (1873), the Czech Photographer Association (1882), the Czech Tourist Club (1888). Czechs constructed the Czech National Theatre (1881) with many renowned Czech artists taking part.
It wasn't until the 20th century when Czechs made decisive efforts to pursue political independence.
Source: Petr Hora-Hořejš, "Toulky českou minulostí", part 8